Is there a math equivalent to the conditional ternary operator?what is ≡ operator equal to in math?Math...

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Is there a math equivalent to the conditional ternary operator?

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Is there a math equivalent to the conditional ternary operator?


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Is there a math equivalent of the ternary conditional operator as used in programming?



a = b + (c > 0 ? 1 : 2)


The above means that if $c$ is greater than $0$ then $a = b + 1$, otherwise $a = b + 2$.










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  • 4




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    @Alex $a = b + 2 - u(c)$
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    – eyeballfrog
    yesterday








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    A better question here is: what is good notation in this case. There are many ways of writing this, many ways which one would almost never use (like the accepted answer).
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    – Winther
    yesterday








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    In C-derived languages at least, you have made a grievous error! Your expression parses as a = (b + c) > 0 ? 1 : 2. I always use parentheses in these cases, even when they are not strictly necessary.
    $endgroup$
    – TonyK
    yesterday








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    Just to clarify, by “ternary operator” you really mean “conditional operator”? Or are interested in any kind of ternary operator, regardless of semantics?
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    – Konrad Rudolph
    yesterday






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    $begingroup$
    What @KonradRudolph said; you are showing the conditional operator, which is a ternary (takes three arguments; compare unary and binary) operator. I don't think the C language has any other ternary operator, and some form of the conditional operator is found in many languages because it's very convenient to have, but there's no rule that says a language couldn't include other ternary operators alongside (or without having) a conditional operator.
    $endgroup$
    – a CVn
    yesterday
















38












$begingroup$


Is there a math equivalent of the ternary conditional operator as used in programming?



a = b + (c > 0 ? 1 : 2)


The above means that if $c$ is greater than $0$ then $a = b + 1$, otherwise $a = b + 2$.










share|cite|improve this question











$endgroup$








  • 4




    $begingroup$
    @Alex $a = b + 2 - u(c)$
    $endgroup$
    – eyeballfrog
    yesterday








  • 14




    $begingroup$
    A better question here is: what is good notation in this case. There are many ways of writing this, many ways which one would almost never use (like the accepted answer).
    $endgroup$
    – Winther
    yesterday








  • 5




    $begingroup$
    In C-derived languages at least, you have made a grievous error! Your expression parses as a = (b + c) > 0 ? 1 : 2. I always use parentheses in these cases, even when they are not strictly necessary.
    $endgroup$
    – TonyK
    yesterday








  • 3




    $begingroup$
    Just to clarify, by “ternary operator” you really mean “conditional operator”? Or are interested in any kind of ternary operator, regardless of semantics?
    $endgroup$
    – Konrad Rudolph
    yesterday






  • 1




    $begingroup$
    What @KonradRudolph said; you are showing the conditional operator, which is a ternary (takes three arguments; compare unary and binary) operator. I don't think the C language has any other ternary operator, and some form of the conditional operator is found in many languages because it's very convenient to have, but there's no rule that says a language couldn't include other ternary operators alongside (or without having) a conditional operator.
    $endgroup$
    – a CVn
    yesterday














38












38








38


7



$begingroup$


Is there a math equivalent of the ternary conditional operator as used in programming?



a = b + (c > 0 ? 1 : 2)


The above means that if $c$ is greater than $0$ then $a = b + 1$, otherwise $a = b + 2$.










share|cite|improve this question











$endgroup$




Is there a math equivalent of the ternary conditional operator as used in programming?



a = b + (c > 0 ? 1 : 2)


The above means that if $c$ is greater than $0$ then $a = b + 1$, otherwise $a = b + 2$.







notation computer-science






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edited 7 hours ago









PJTraill

683519




683519










asked yesterday









dataphiledataphile

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  • 4




    $begingroup$
    @Alex $a = b + 2 - u(c)$
    $endgroup$
    – eyeballfrog
    yesterday








  • 14




    $begingroup$
    A better question here is: what is good notation in this case. There are many ways of writing this, many ways which one would almost never use (like the accepted answer).
    $endgroup$
    – Winther
    yesterday








  • 5




    $begingroup$
    In C-derived languages at least, you have made a grievous error! Your expression parses as a = (b + c) > 0 ? 1 : 2. I always use parentheses in these cases, even when they are not strictly necessary.
    $endgroup$
    – TonyK
    yesterday








  • 3




    $begingroup$
    Just to clarify, by “ternary operator” you really mean “conditional operator”? Or are interested in any kind of ternary operator, regardless of semantics?
    $endgroup$
    – Konrad Rudolph
    yesterday






  • 1




    $begingroup$
    What @KonradRudolph said; you are showing the conditional operator, which is a ternary (takes three arguments; compare unary and binary) operator. I don't think the C language has any other ternary operator, and some form of the conditional operator is found in many languages because it's very convenient to have, but there's no rule that says a language couldn't include other ternary operators alongside (or without having) a conditional operator.
    $endgroup$
    – a CVn
    yesterday














  • 4




    $begingroup$
    @Alex $a = b + 2 - u(c)$
    $endgroup$
    – eyeballfrog
    yesterday








  • 14




    $begingroup$
    A better question here is: what is good notation in this case. There are many ways of writing this, many ways which one would almost never use (like the accepted answer).
    $endgroup$
    – Winther
    yesterday








  • 5




    $begingroup$
    In C-derived languages at least, you have made a grievous error! Your expression parses as a = (b + c) > 0 ? 1 : 2. I always use parentheses in these cases, even when they are not strictly necessary.
    $endgroup$
    – TonyK
    yesterday








  • 3




    $begingroup$
    Just to clarify, by “ternary operator” you really mean “conditional operator”? Or are interested in any kind of ternary operator, regardless of semantics?
    $endgroup$
    – Konrad Rudolph
    yesterday






  • 1




    $begingroup$
    What @KonradRudolph said; you are showing the conditional operator, which is a ternary (takes three arguments; compare unary and binary) operator. I don't think the C language has any other ternary operator, and some form of the conditional operator is found in many languages because it's very convenient to have, but there's no rule that says a language couldn't include other ternary operators alongside (or without having) a conditional operator.
    $endgroup$
    – a CVn
    yesterday








4




4




$begingroup$
@Alex $a = b + 2 - u(c)$
$endgroup$
– eyeballfrog
yesterday






$begingroup$
@Alex $a = b + 2 - u(c)$
$endgroup$
– eyeballfrog
yesterday






14




14




$begingroup$
A better question here is: what is good notation in this case. There are many ways of writing this, many ways which one would almost never use (like the accepted answer).
$endgroup$
– Winther
yesterday






$begingroup$
A better question here is: what is good notation in this case. There are many ways of writing this, many ways which one would almost never use (like the accepted answer).
$endgroup$
– Winther
yesterday






5




5




$begingroup$
In C-derived languages at least, you have made a grievous error! Your expression parses as a = (b + c) > 0 ? 1 : 2. I always use parentheses in these cases, even when they are not strictly necessary.
$endgroup$
– TonyK
yesterday






$begingroup$
In C-derived languages at least, you have made a grievous error! Your expression parses as a = (b + c) > 0 ? 1 : 2. I always use parentheses in these cases, even when they are not strictly necessary.
$endgroup$
– TonyK
yesterday






3




3




$begingroup$
Just to clarify, by “ternary operator” you really mean “conditional operator”? Or are interested in any kind of ternary operator, regardless of semantics?
$endgroup$
– Konrad Rudolph
yesterday




$begingroup$
Just to clarify, by “ternary operator” you really mean “conditional operator”? Or are interested in any kind of ternary operator, regardless of semantics?
$endgroup$
– Konrad Rudolph
yesterday




1




1




$begingroup$
What @KonradRudolph said; you are showing the conditional operator, which is a ternary (takes three arguments; compare unary and binary) operator. I don't think the C language has any other ternary operator, and some form of the conditional operator is found in many languages because it's very convenient to have, but there's no rule that says a language couldn't include other ternary operators alongside (or without having) a conditional operator.
$endgroup$
– a CVn
yesterday




$begingroup$
What @KonradRudolph said; you are showing the conditional operator, which is a ternary (takes three arguments; compare unary and binary) operator. I don't think the C language has any other ternary operator, and some form of the conditional operator is found in many languages because it's very convenient to have, but there's no rule that says a language couldn't include other ternary operators alongside (or without having) a conditional operator.
$endgroup$
– a CVn
yesterday










10 Answers
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From physics, I'm used to seeing the Kronecker delta,$$
{delta}_{ij}
equiv
left{
begin{array}{lll}
1 &text{if} & i=j \
0 &text{else}
end{array}
right. _{,}
$$
and I think people who work with it find the slightly generalized notation$$
{delta}_{left[text{condition}right]}
equiv
left{
begin{array}{lll}
1 &text{if} & left[text{condition}right] \
0 &text{else}
end{array}
right.
$$
to be pretty natural to them.



So, I tend to use $delta_{left[text{condition}right]}$ for a lot of things. Just seems so simple and well-understood.



Transforms:




  1. Basic Kronecker delta:
    To write the basic Kronecker delta in terms of the generalized Kronecker delta, it's just$$
    delta_{ij}
    Rightarrow
    delta_{i=j}
    ,.$$
    It's almost the same notation, and I think most folks can figure it out pretty easily without needing it explained.


  2. Conditional operator:
    The "conditional operator" or "ternary operator" for the simple case of ?1:0:$$
    begin{array}{ccccc}
    boxed{
    begin{array}{l}
    texttt{if}~left(texttt{condition}right) \
    { \
    ~~~~texttt{return 1;} \
    } \
    texttt{else} \
    { \
    ~~~~texttt{return 0;} \
    }
    end{array}
    ~} &
    Rightarrow &
    boxed{~
    texttt{condition ? 1 : 0}
    ~} &
    Rightarrow &
    delta_{i=j}
    end{array}
    _{.}
    $$
    Then if you want a non-zero value for the false-case, you'd just add another Kronecker delta, $delta_{operatorname{NOT}left(left[text{condition}right]right)} ,$ e.g. $delta_{i neq j} .$


  3. Indicator function:
    @SiongThyeGoh's answer recommended using indicator function notation. I'd rewrite their example like$$
    begin{array}{ccccc}
    underbrace{a=b+1+mathbb{1}_{(-infty, 0]}(c)}
    _{text{their example}}
    &
    Rightarrow &
    underbrace{a=b+1+ delta_{c in left(-infty, 0right]}}
    _{text{direct translation}} &
    Rightarrow &
    underbrace{a=b+1+ delta_{c , {small{leq}} , 0}}
    _{text{cleaner form}}
    end{array}
    ,.
    $$


  4. Iverson bracket:
    Iverson bracket notation, as suggested in @FredH's answer, is apparently the same thing; according to Wikipedia, it's meant as a generalization of the Kronecker delta, except they drop the $delta$ entirely, just putting the condition in square-brackets. In a context in readers expect it, Iverson bracket notation might be preferable if conditionals will be used a lot.





Note: "Conditional operator" rather than "ternary operator".



The conditional operator, condition ? trueValue : falseValue, has 3 arguments, making it an example of a ternary operator. By contrast, most other operators in programming tend to be unary operators (which have 1 argument) or binary operators (which have 2 arguments).



Since the conditional operator is fairly unique in being a ternary operator, it's often been called "the ternary operator", leading many to believe that that's its name. However, "conditional operator" is more specific and should generally be preferred.






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    wow, this is beautiful. I need it for an infinite series (with some elements excluded i.e. multiplied by zero and others included, multiplied by 1) so this cleaner form is exactly what I need.
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    – dataphile
    yesterday










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    @dataphile Yeah, the Kronecker delta's great for specifying elements like that; it comes in handy if you're using Sigma/Pi/Einstein notations. Worth noting that it's basically the integrated Dirac delta; if you're working in a continuous domain, then the Dirac delta tends to be preferred, though it's largely the same thing.
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    – Nat
    yesterday








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    The Kronecker delta isn’t a ternary operator, it’s a plain old binary operator, since it has two operands.
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    – Konrad Rudolph
    yesterday












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    @KonradRudolph Yeah, gotta add a $delta_{operatorname{NOT}left(left[text{condition}right]right)}$-term to get that false-case behavior in there when it's non-zero.
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    – Nat
    yesterday








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    The $1$-if-condition-else-$0$ generalisation of the Kronecker delta is called the Iverson bracket. We can generalise to any two input values we like with a suitable function, of domain ${0,,1}$, wraping the Iverson bracket.
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    – J.G.
    yesterday



















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The expression b + (c > 0 ? 1 : 2) is not a ternary operator; it is a function of two variables. There is one operation that results in $a$. You can certainly define a function
$$f(b,c)=begin {cases} b+1&c gt 0\
b+2 & c le 0 end {cases}$$



You can also define functions with any number of inputs you want, so you can define $f(a,b,c)=a(b+c^2)$, for example. This is a ternary function.






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    Ternary operator in programming means a conditional statement of form "if A then x else y" and uually is written just as presented in the OP, i.e. A ? x : y. It would also literally be termary if the OP wouldn't have inserted the particular values of 1 and 2.
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    – Džuris
    yesterday






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    @Džuris Ternary means "of arity 3", like binary means "of arity 2" and unary "of arity 1". The specific operator is the conditional ternary operator.
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    – frabala
    yesterday






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    Once again, clarity beats compactness when it comes to mathematics.
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    – Asaf Karagila
    yesterday






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    Well-chosen compact notations can contribute to clarity rather than detract from it, though.
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    – Henning Makholm
    yesterday






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    @Džuris No it isn’t. The link you cite says, in its very first sentence, that “?: is a ternary operator” (not “the ternary operator”; emphasis mine). Some people refer to it as “the ternary operator” but this is strictly incorrect (and merits a correction e.g. on Stack Overflow).
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    – Konrad Rudolph
    yesterday





















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In Concrete Mathematics by Graham, Knuth and Patashnik, the authors use the "Iverson bracket" notation: Square brackets around a statement represent $1$ if the statement is true and $0$ otherwise. Using this notation, you could write
$$
a = b + 2 - [c gt 0].
$$






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    aka en.wikipedia.org/wiki/Iverson_bracket
    $endgroup$
    – qwr
    yesterday










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    @qwr Thank you! I did not recall the name.
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    – FredH
    yesterday






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    I would find it clearer to write $a = b + 1[c>0] + 2[cle0]$.
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    – Rahul
    yesterday








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    I personally find Iverson brackets less cluttered than indicator functions, so I definitely recommend this notation.
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    – J. M. is not a mathematician
    yesterday






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    Surely this is a unary operator, which works much like for example ! in the C programming language: Input is an arbitrary expression, the result is either 0 or 1 depending on the expression.
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    – pipe
    yesterday





















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Using the indicator function notation:$$a=b+1+mathbb{1}_{(-infty, 0]}(c)$$






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    Indicator is definitely the way to go, since the conditional can define an arbitrary set.
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    – eyeballfrog
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    Thank you, that is very elegant. I will definitely try out all of the answers as I often use this in various scenarios.
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    – dataphile
    yesterday






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    I often use - and have seen others use - $mathbb{1}$ ("blackboard bold") to denote the indicator function to distinguish it from the number $1$
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    – Christopher
    yesterday








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    The indicator function isn’t a ternary operator, it only has two operands.
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    – Konrad Rudolph
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In math, equations are written in piecewise form by having a curly brace enclose multiple lines; each one with a condition excepting the last which has "otherwise".



There are a few custom operators that also occasionally make an appearance. E.g. the Heavyside function mentioned by Alex, the Dirac function, and the cyclical operator $delta_{ijk}$ - all of which can be used to emulate conditional behaviour.






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    One should realize that operators are just a fancy way of using functions.
    So a ternary operator is a function of 3 variables that is notated in a different way. Is that useful? The answer is mostly not. Also realize that any mathematician is allowed to introduce any notation he feels is illustrative.



    Let's review why we use binary operators at all like in a+b*c . Because parameters and results are of the same type, it makes sense to leave out parentheses and introduce complicated priority rules. Imagine that a b c are numbers and we have a normal + and a peculiar * that results in dragons. Now the expression doesn't make sense (assumming a high priority *), because there is no way to add numbers and dragons. Thusly most ternary operators results in a mess.



    With a proper notation there are examples of ternary operations. For example, there is a special notation for "sum for i from a to b of expression". This takes two boundaries (numbers) and a function from a number of that type that results in another number. (Mathematician, read "element of an addition group" for number.)
    The notation for integration is similarly ternary.

    So in short ternary operators exist, and you can define your own. They are in general accompagnied with a special notation, or they are not helpful.



    Now back to the special case you mention.
    Because truth values are implied in math, an expression like "if a then b else c" makes sense if a represens a truth value like (7<12). The above expression is understood in every mathematical context. However in a context where truth values are not considered a set, (if .. then .. else ..) would not be considered an operator/function, but a textual explanation. A general accepted notation could be useful in math, but I'm not aware there is one. That is probably, because like in the above, informal notations are readily understood.






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      Following solution is not defined for $c = 0$; however it uses very basic operations only, which might be useful as you probably look for an expression to implement in a program:



      $$a = b + 1lambda + 2(1-lambda)$$



      where



      $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$





      You need to make the problem discrete and make a choice from two values. So, given some value $c in mathbb{R}$ we need to calculate some value $lambda in {0,1}$ depnding on $c<0$ or $c>0$.



      Knowing that



      $$frac{|c|}{c} in {1,-1}$$



      we can calculate the $lambda$ as follows:



      $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$



      Now that our $lambda in {0,1}$ we can do the "choice" between the two constants $d$ and $e$ as follows:



      $$dlambda + e(1-lambda)$$



      which equals $d$ for $lambda = 1$, and $e$ for $lambda = 0$.






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      New contributor




      Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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        $begingroup$

        There are many good answers that give notation for, “if this condition holds, then 1, else 0.” This corresponds to an even simpler expression in C;(x>1) is equivalent to (x>1 ? 1 : 0).



        It’s worth noting that the ternary operator is more general than that. If the arguments are elements of a ring, you could express c ? a : b with (using Iverson-bracket notation) $(a-b) cdot [c] + b$, but not otherwise. (And compilers frequently use this trick, in a Boolean ring, to compile conditionals without needing to execute a branch instruction.) In a C program, evaluating the expressions $a$ or $b$ might have side-effects, such as deleting a file or printing a message to the screen. In a mathematical function, this isn’t something you would worry about, and a programming language where this is impossible is called functional.



        Ross Millikan gave the most standard notation, a cases block. The closest equivalent in mathematical computer science is the if-then-else function of Lambda Calculus.






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        • $begingroup$
          "in C;(x>1) is equivalent to (x>1 ? 1 : 0)" Not exactly. Rather, C considers any nonzero value as equivalent from a truthiness perspective. So there is no difference between an integer expression taking the value of -1, 1, 42 or INT_MAX when that expression is treated as a boolean rvalue. In C, the one special integer value, when treated as a boolean, is 0, representing false. That said, if someone actually used (x>1) as a non-boolean expression and I noticed it, I would likely at least briefly try to find some heavy physical object that could be applied at high speed to their keyboard.
          $endgroup$
          – a CVn
          yesterday






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          $begingroup$
          @aCVn The C11 standard states, “Each of the operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) shall yield 1 if the specified relation is true and 0 if it is false.” Similarly for equality and inequality, “Each of the operators yields 1 if the specified relation is true and 0 if it is false.”
          $endgroup$
          – Davislor
          yesterday












        • $begingroup$
          @aCVn What you wrote is true, but the relational operators in C will only return the values 1 or 0.
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          – Davislor
          yesterday



















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        I think mathematicians should not be afraid to use the Iverson bracket, including when teaching, as this is a generally very useful notation whose only slightly unusual feature is to introduce a logical expression in the middle of an algebraic one (but one already regularly finds conditions inside set-theoretic expressions, so it really is not a big deal). It may avoid a lot of clutter, notably many instances of clumsy expressions by cases with a big unmatched brace (which is usually only usable as the right hand side of a definition). Since brackets do have many other uses in mathematics, I personally prefer a typographically distinct representation of Iverson brackets, rendering your example as
        $$def[#1]{[![{#1}]!]}
        a= b + [c>0]1 + [cnot>0]2.
        $$

        This works best in additive context (though one can use Iverson brackets in the exponent for optional multiplicative factors). It is not really ideal for general two-way branches, as the condition must be repeated twice, one of them in negated form, but it happens that most of the time one needs $0$ as the value for one branch anyway.



        As a more concise two-way branch, I can recall that Algol68 introduced the notation $b+(c>0mid 1mid 2)$ for the right-hand side of your equation; though this is a programming language and not mathematics, it was designed by mathematicians. They also had notation for multi-way branching: thus the solution to the recursion $a_{n+2}=a_{n+1}-a_n$ with initial values $a_0=0$, $a_1=1$ can be written
        $$
        a_n=(nbmod 6+1mid 0,1,1,0,-1,-1)
        $$

        (where the "${}+1$" is needed because in 1968 they still counted starting from $1$, which is a mistake), which is reasonably concise and readable, compared to other ways to express this result. Also consider, for month $m$ in year $y$, the number
        $$
        ( m mid 31,(ybmod 4=0land ybmod 100neq0lor ybmod400=0mid 29mid 28)
        ,31,30,31,30,31,31,30,31,30,31).
        $$






        share|cite|improve this answer









        $endgroup$





















          0












          $begingroup$

          It can be viewed as inline function:



          $$f(c)=begin {cases} 1&c gt 0\
          2& c le 0 end {cases}$$



          so it comes as:



          $$a = b + f(c)$$
          used in everyday mathematics.






          share|cite|improve this answer









          $endgroup$













            Your Answer





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            10 Answers
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            active

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            10 Answers
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            active

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            active

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            active

            oldest

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            49












            $begingroup$

            From physics, I'm used to seeing the Kronecker delta,$$
            {delta}_{ij}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & i=j \
            0 &text{else}
            end{array}
            right. _{,}
            $$
            and I think people who work with it find the slightly generalized notation$$
            {delta}_{left[text{condition}right]}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & left[text{condition}right] \
            0 &text{else}
            end{array}
            right.
            $$
            to be pretty natural to them.



            So, I tend to use $delta_{left[text{condition}right]}$ for a lot of things. Just seems so simple and well-understood.



            Transforms:




            1. Basic Kronecker delta:
              To write the basic Kronecker delta in terms of the generalized Kronecker delta, it's just$$
              delta_{ij}
              Rightarrow
              delta_{i=j}
              ,.$$
              It's almost the same notation, and I think most folks can figure it out pretty easily without needing it explained.


            2. Conditional operator:
              The "conditional operator" or "ternary operator" for the simple case of ?1:0:$$
              begin{array}{ccccc}
              boxed{
              begin{array}{l}
              texttt{if}~left(texttt{condition}right) \
              { \
              ~~~~texttt{return 1;} \
              } \
              texttt{else} \
              { \
              ~~~~texttt{return 0;} \
              }
              end{array}
              ~} &
              Rightarrow &
              boxed{~
              texttt{condition ? 1 : 0}
              ~} &
              Rightarrow &
              delta_{i=j}
              end{array}
              _{.}
              $$
              Then if you want a non-zero value for the false-case, you'd just add another Kronecker delta, $delta_{operatorname{NOT}left(left[text{condition}right]right)} ,$ e.g. $delta_{i neq j} .$


            3. Indicator function:
              @SiongThyeGoh's answer recommended using indicator function notation. I'd rewrite their example like$$
              begin{array}{ccccc}
              underbrace{a=b+1+mathbb{1}_{(-infty, 0]}(c)}
              _{text{their example}}
              &
              Rightarrow &
              underbrace{a=b+1+ delta_{c in left(-infty, 0right]}}
              _{text{direct translation}} &
              Rightarrow &
              underbrace{a=b+1+ delta_{c , {small{leq}} , 0}}
              _{text{cleaner form}}
              end{array}
              ,.
              $$


            4. Iverson bracket:
              Iverson bracket notation, as suggested in @FredH's answer, is apparently the same thing; according to Wikipedia, it's meant as a generalization of the Kronecker delta, except they drop the $delta$ entirely, just putting the condition in square-brackets. In a context in readers expect it, Iverson bracket notation might be preferable if conditionals will be used a lot.





            Note: "Conditional operator" rather than "ternary operator".



            The conditional operator, condition ? trueValue : falseValue, has 3 arguments, making it an example of a ternary operator. By contrast, most other operators in programming tend to be unary operators (which have 1 argument) or binary operators (which have 2 arguments).



            Since the conditional operator is fairly unique in being a ternary operator, it's often been called "the ternary operator", leading many to believe that that's its name. However, "conditional operator" is more specific and should generally be preferred.






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              wow, this is beautiful. I need it for an infinite series (with some elements excluded i.e. multiplied by zero and others included, multiplied by 1) so this cleaner form is exactly what I need.
              $endgroup$
              – dataphile
              yesterday










            • $begingroup$
              @dataphile Yeah, the Kronecker delta's great for specifying elements like that; it comes in handy if you're using Sigma/Pi/Einstein notations. Worth noting that it's basically the integrated Dirac delta; if you're working in a continuous domain, then the Dirac delta tends to be preferred, though it's largely the same thing.
              $endgroup$
              – Nat
              yesterday








            • 2




              $begingroup$
              The Kronecker delta isn’t a ternary operator, it’s a plain old binary operator, since it has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday












            • $begingroup$
              @KonradRudolph Yeah, gotta add a $delta_{operatorname{NOT}left(left[text{condition}right]right)}$-term to get that false-case behavior in there when it's non-zero.
              $endgroup$
              – Nat
              yesterday








            • 1




              $begingroup$
              The $1$-if-condition-else-$0$ generalisation of the Kronecker delta is called the Iverson bracket. We can generalise to any two input values we like with a suitable function, of domain ${0,,1}$, wraping the Iverson bracket.
              $endgroup$
              – J.G.
              yesterday
















            49












            $begingroup$

            From physics, I'm used to seeing the Kronecker delta,$$
            {delta}_{ij}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & i=j \
            0 &text{else}
            end{array}
            right. _{,}
            $$
            and I think people who work with it find the slightly generalized notation$$
            {delta}_{left[text{condition}right]}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & left[text{condition}right] \
            0 &text{else}
            end{array}
            right.
            $$
            to be pretty natural to them.



            So, I tend to use $delta_{left[text{condition}right]}$ for a lot of things. Just seems so simple and well-understood.



            Transforms:




            1. Basic Kronecker delta:
              To write the basic Kronecker delta in terms of the generalized Kronecker delta, it's just$$
              delta_{ij}
              Rightarrow
              delta_{i=j}
              ,.$$
              It's almost the same notation, and I think most folks can figure it out pretty easily without needing it explained.


            2. Conditional operator:
              The "conditional operator" or "ternary operator" for the simple case of ?1:0:$$
              begin{array}{ccccc}
              boxed{
              begin{array}{l}
              texttt{if}~left(texttt{condition}right) \
              { \
              ~~~~texttt{return 1;} \
              } \
              texttt{else} \
              { \
              ~~~~texttt{return 0;} \
              }
              end{array}
              ~} &
              Rightarrow &
              boxed{~
              texttt{condition ? 1 : 0}
              ~} &
              Rightarrow &
              delta_{i=j}
              end{array}
              _{.}
              $$
              Then if you want a non-zero value for the false-case, you'd just add another Kronecker delta, $delta_{operatorname{NOT}left(left[text{condition}right]right)} ,$ e.g. $delta_{i neq j} .$


            3. Indicator function:
              @SiongThyeGoh's answer recommended using indicator function notation. I'd rewrite their example like$$
              begin{array}{ccccc}
              underbrace{a=b+1+mathbb{1}_{(-infty, 0]}(c)}
              _{text{their example}}
              &
              Rightarrow &
              underbrace{a=b+1+ delta_{c in left(-infty, 0right]}}
              _{text{direct translation}} &
              Rightarrow &
              underbrace{a=b+1+ delta_{c , {small{leq}} , 0}}
              _{text{cleaner form}}
              end{array}
              ,.
              $$


            4. Iverson bracket:
              Iverson bracket notation, as suggested in @FredH's answer, is apparently the same thing; according to Wikipedia, it's meant as a generalization of the Kronecker delta, except they drop the $delta$ entirely, just putting the condition in square-brackets. In a context in readers expect it, Iverson bracket notation might be preferable if conditionals will be used a lot.





            Note: "Conditional operator" rather than "ternary operator".



            The conditional operator, condition ? trueValue : falseValue, has 3 arguments, making it an example of a ternary operator. By contrast, most other operators in programming tend to be unary operators (which have 1 argument) or binary operators (which have 2 arguments).



            Since the conditional operator is fairly unique in being a ternary operator, it's often been called "the ternary operator", leading many to believe that that's its name. However, "conditional operator" is more specific and should generally be preferred.






            share|cite|improve this answer











            $endgroup$













            • $begingroup$
              wow, this is beautiful. I need it for an infinite series (with some elements excluded i.e. multiplied by zero and others included, multiplied by 1) so this cleaner form is exactly what I need.
              $endgroup$
              – dataphile
              yesterday










            • $begingroup$
              @dataphile Yeah, the Kronecker delta's great for specifying elements like that; it comes in handy if you're using Sigma/Pi/Einstein notations. Worth noting that it's basically the integrated Dirac delta; if you're working in a continuous domain, then the Dirac delta tends to be preferred, though it's largely the same thing.
              $endgroup$
              – Nat
              yesterday








            • 2




              $begingroup$
              The Kronecker delta isn’t a ternary operator, it’s a plain old binary operator, since it has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday












            • $begingroup$
              @KonradRudolph Yeah, gotta add a $delta_{operatorname{NOT}left(left[text{condition}right]right)}$-term to get that false-case behavior in there when it's non-zero.
              $endgroup$
              – Nat
              yesterday








            • 1




              $begingroup$
              The $1$-if-condition-else-$0$ generalisation of the Kronecker delta is called the Iverson bracket. We can generalise to any two input values we like with a suitable function, of domain ${0,,1}$, wraping the Iverson bracket.
              $endgroup$
              – J.G.
              yesterday














            49












            49








            49





            $begingroup$

            From physics, I'm used to seeing the Kronecker delta,$$
            {delta}_{ij}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & i=j \
            0 &text{else}
            end{array}
            right. _{,}
            $$
            and I think people who work with it find the slightly generalized notation$$
            {delta}_{left[text{condition}right]}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & left[text{condition}right] \
            0 &text{else}
            end{array}
            right.
            $$
            to be pretty natural to them.



            So, I tend to use $delta_{left[text{condition}right]}$ for a lot of things. Just seems so simple and well-understood.



            Transforms:




            1. Basic Kronecker delta:
              To write the basic Kronecker delta in terms of the generalized Kronecker delta, it's just$$
              delta_{ij}
              Rightarrow
              delta_{i=j}
              ,.$$
              It's almost the same notation, and I think most folks can figure it out pretty easily without needing it explained.


            2. Conditional operator:
              The "conditional operator" or "ternary operator" for the simple case of ?1:0:$$
              begin{array}{ccccc}
              boxed{
              begin{array}{l}
              texttt{if}~left(texttt{condition}right) \
              { \
              ~~~~texttt{return 1;} \
              } \
              texttt{else} \
              { \
              ~~~~texttt{return 0;} \
              }
              end{array}
              ~} &
              Rightarrow &
              boxed{~
              texttt{condition ? 1 : 0}
              ~} &
              Rightarrow &
              delta_{i=j}
              end{array}
              _{.}
              $$
              Then if you want a non-zero value for the false-case, you'd just add another Kronecker delta, $delta_{operatorname{NOT}left(left[text{condition}right]right)} ,$ e.g. $delta_{i neq j} .$


            3. Indicator function:
              @SiongThyeGoh's answer recommended using indicator function notation. I'd rewrite their example like$$
              begin{array}{ccccc}
              underbrace{a=b+1+mathbb{1}_{(-infty, 0]}(c)}
              _{text{their example}}
              &
              Rightarrow &
              underbrace{a=b+1+ delta_{c in left(-infty, 0right]}}
              _{text{direct translation}} &
              Rightarrow &
              underbrace{a=b+1+ delta_{c , {small{leq}} , 0}}
              _{text{cleaner form}}
              end{array}
              ,.
              $$


            4. Iverson bracket:
              Iverson bracket notation, as suggested in @FredH's answer, is apparently the same thing; according to Wikipedia, it's meant as a generalization of the Kronecker delta, except they drop the $delta$ entirely, just putting the condition in square-brackets. In a context in readers expect it, Iverson bracket notation might be preferable if conditionals will be used a lot.





            Note: "Conditional operator" rather than "ternary operator".



            The conditional operator, condition ? trueValue : falseValue, has 3 arguments, making it an example of a ternary operator. By contrast, most other operators in programming tend to be unary operators (which have 1 argument) or binary operators (which have 2 arguments).



            Since the conditional operator is fairly unique in being a ternary operator, it's often been called "the ternary operator", leading many to believe that that's its name. However, "conditional operator" is more specific and should generally be preferred.






            share|cite|improve this answer











            $endgroup$



            From physics, I'm used to seeing the Kronecker delta,$$
            {delta}_{ij}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & i=j \
            0 &text{else}
            end{array}
            right. _{,}
            $$
            and I think people who work with it find the slightly generalized notation$$
            {delta}_{left[text{condition}right]}
            equiv
            left{
            begin{array}{lll}
            1 &text{if} & left[text{condition}right] \
            0 &text{else}
            end{array}
            right.
            $$
            to be pretty natural to them.



            So, I tend to use $delta_{left[text{condition}right]}$ for a lot of things. Just seems so simple and well-understood.



            Transforms:




            1. Basic Kronecker delta:
              To write the basic Kronecker delta in terms of the generalized Kronecker delta, it's just$$
              delta_{ij}
              Rightarrow
              delta_{i=j}
              ,.$$
              It's almost the same notation, and I think most folks can figure it out pretty easily without needing it explained.


            2. Conditional operator:
              The "conditional operator" or "ternary operator" for the simple case of ?1:0:$$
              begin{array}{ccccc}
              boxed{
              begin{array}{l}
              texttt{if}~left(texttt{condition}right) \
              { \
              ~~~~texttt{return 1;} \
              } \
              texttt{else} \
              { \
              ~~~~texttt{return 0;} \
              }
              end{array}
              ~} &
              Rightarrow &
              boxed{~
              texttt{condition ? 1 : 0}
              ~} &
              Rightarrow &
              delta_{i=j}
              end{array}
              _{.}
              $$
              Then if you want a non-zero value for the false-case, you'd just add another Kronecker delta, $delta_{operatorname{NOT}left(left[text{condition}right]right)} ,$ e.g. $delta_{i neq j} .$


            3. Indicator function:
              @SiongThyeGoh's answer recommended using indicator function notation. I'd rewrite their example like$$
              begin{array}{ccccc}
              underbrace{a=b+1+mathbb{1}_{(-infty, 0]}(c)}
              _{text{their example}}
              &
              Rightarrow &
              underbrace{a=b+1+ delta_{c in left(-infty, 0right]}}
              _{text{direct translation}} &
              Rightarrow &
              underbrace{a=b+1+ delta_{c , {small{leq}} , 0}}
              _{text{cleaner form}}
              end{array}
              ,.
              $$


            4. Iverson bracket:
              Iverson bracket notation, as suggested in @FredH's answer, is apparently the same thing; according to Wikipedia, it's meant as a generalization of the Kronecker delta, except they drop the $delta$ entirely, just putting the condition in square-brackets. In a context in readers expect it, Iverson bracket notation might be preferable if conditionals will be used a lot.





            Note: "Conditional operator" rather than "ternary operator".



            The conditional operator, condition ? trueValue : falseValue, has 3 arguments, making it an example of a ternary operator. By contrast, most other operators in programming tend to be unary operators (which have 1 argument) or binary operators (which have 2 arguments).



            Since the conditional operator is fairly unique in being a ternary operator, it's often been called "the ternary operator", leading many to believe that that's its name. However, "conditional operator" is more specific and should generally be preferred.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited yesterday

























            answered yesterday









            NatNat

            9581713




            9581713












            • $begingroup$
              wow, this is beautiful. I need it for an infinite series (with some elements excluded i.e. multiplied by zero and others included, multiplied by 1) so this cleaner form is exactly what I need.
              $endgroup$
              – dataphile
              yesterday










            • $begingroup$
              @dataphile Yeah, the Kronecker delta's great for specifying elements like that; it comes in handy if you're using Sigma/Pi/Einstein notations. Worth noting that it's basically the integrated Dirac delta; if you're working in a continuous domain, then the Dirac delta tends to be preferred, though it's largely the same thing.
              $endgroup$
              – Nat
              yesterday








            • 2




              $begingroup$
              The Kronecker delta isn’t a ternary operator, it’s a plain old binary operator, since it has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday












            • $begingroup$
              @KonradRudolph Yeah, gotta add a $delta_{operatorname{NOT}left(left[text{condition}right]right)}$-term to get that false-case behavior in there when it's non-zero.
              $endgroup$
              – Nat
              yesterday








            • 1




              $begingroup$
              The $1$-if-condition-else-$0$ generalisation of the Kronecker delta is called the Iverson bracket. We can generalise to any two input values we like with a suitable function, of domain ${0,,1}$, wraping the Iverson bracket.
              $endgroup$
              – J.G.
              yesterday


















            • $begingroup$
              wow, this is beautiful. I need it for an infinite series (with some elements excluded i.e. multiplied by zero and others included, multiplied by 1) so this cleaner form is exactly what I need.
              $endgroup$
              – dataphile
              yesterday










            • $begingroup$
              @dataphile Yeah, the Kronecker delta's great for specifying elements like that; it comes in handy if you're using Sigma/Pi/Einstein notations. Worth noting that it's basically the integrated Dirac delta; if you're working in a continuous domain, then the Dirac delta tends to be preferred, though it's largely the same thing.
              $endgroup$
              – Nat
              yesterday








            • 2




              $begingroup$
              The Kronecker delta isn’t a ternary operator, it’s a plain old binary operator, since it has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday












            • $begingroup$
              @KonradRudolph Yeah, gotta add a $delta_{operatorname{NOT}left(left[text{condition}right]right)}$-term to get that false-case behavior in there when it's non-zero.
              $endgroup$
              – Nat
              yesterday








            • 1




              $begingroup$
              The $1$-if-condition-else-$0$ generalisation of the Kronecker delta is called the Iverson bracket. We can generalise to any two input values we like with a suitable function, of domain ${0,,1}$, wraping the Iverson bracket.
              $endgroup$
              – J.G.
              yesterday
















            $begingroup$
            wow, this is beautiful. I need it for an infinite series (with some elements excluded i.e. multiplied by zero and others included, multiplied by 1) so this cleaner form is exactly what I need.
            $endgroup$
            – dataphile
            yesterday




            $begingroup$
            wow, this is beautiful. I need it for an infinite series (with some elements excluded i.e. multiplied by zero and others included, multiplied by 1) so this cleaner form is exactly what I need.
            $endgroup$
            – dataphile
            yesterday












            $begingroup$
            @dataphile Yeah, the Kronecker delta's great for specifying elements like that; it comes in handy if you're using Sigma/Pi/Einstein notations. Worth noting that it's basically the integrated Dirac delta; if you're working in a continuous domain, then the Dirac delta tends to be preferred, though it's largely the same thing.
            $endgroup$
            – Nat
            yesterday






            $begingroup$
            @dataphile Yeah, the Kronecker delta's great for specifying elements like that; it comes in handy if you're using Sigma/Pi/Einstein notations. Worth noting that it's basically the integrated Dirac delta; if you're working in a continuous domain, then the Dirac delta tends to be preferred, though it's largely the same thing.
            $endgroup$
            – Nat
            yesterday






            2




            2




            $begingroup$
            The Kronecker delta isn’t a ternary operator, it’s a plain old binary operator, since it has two operands.
            $endgroup$
            – Konrad Rudolph
            yesterday






            $begingroup$
            The Kronecker delta isn’t a ternary operator, it’s a plain old binary operator, since it has two operands.
            $endgroup$
            – Konrad Rudolph
            yesterday














            $begingroup$
            @KonradRudolph Yeah, gotta add a $delta_{operatorname{NOT}left(left[text{condition}right]right)}$-term to get that false-case behavior in there when it's non-zero.
            $endgroup$
            – Nat
            yesterday






            $begingroup$
            @KonradRudolph Yeah, gotta add a $delta_{operatorname{NOT}left(left[text{condition}right]right)}$-term to get that false-case behavior in there when it's non-zero.
            $endgroup$
            – Nat
            yesterday






            1




            1




            $begingroup$
            The $1$-if-condition-else-$0$ generalisation of the Kronecker delta is called the Iverson bracket. We can generalise to any two input values we like with a suitable function, of domain ${0,,1}$, wraping the Iverson bracket.
            $endgroup$
            – J.G.
            yesterday




            $begingroup$
            The $1$-if-condition-else-$0$ generalisation of the Kronecker delta is called the Iverson bracket. We can generalise to any two input values we like with a suitable function, of domain ${0,,1}$, wraping the Iverson bracket.
            $endgroup$
            – J.G.
            yesterday











            53












            $begingroup$

            The expression b + (c > 0 ? 1 : 2) is not a ternary operator; it is a function of two variables. There is one operation that results in $a$. You can certainly define a function
            $$f(b,c)=begin {cases} b+1&c gt 0\
            b+2 & c le 0 end {cases}$$



            You can also define functions with any number of inputs you want, so you can define $f(a,b,c)=a(b+c^2)$, for example. This is a ternary function.






            share|cite|improve this answer











            $endgroup$









            • 9




              $begingroup$
              Ternary operator in programming means a conditional statement of form "if A then x else y" and uually is written just as presented in the OP, i.e. A ? x : y. It would also literally be termary if the OP wouldn't have inserted the particular values of 1 and 2.
              $endgroup$
              – Džuris
              yesterday






            • 15




              $begingroup$
              @Džuris Ternary means "of arity 3", like binary means "of arity 2" and unary "of arity 1". The specific operator is the conditional ternary operator.
              $endgroup$
              – frabala
              yesterday






            • 15




              $begingroup$
              Once again, clarity beats compactness when it comes to mathematics.
              $endgroup$
              – Asaf Karagila
              yesterday






            • 6




              $begingroup$
              Well-chosen compact notations can contribute to clarity rather than detract from it, though.
              $endgroup$
              – Henning Makholm
              yesterday






            • 7




              $begingroup$
              @Džuris No it isn’t. The link you cite says, in its very first sentence, that “?: is a ternary operator” (not “the ternary operator”; emphasis mine). Some people refer to it as “the ternary operator” but this is strictly incorrect (and merits a correction e.g. on Stack Overflow).
              $endgroup$
              – Konrad Rudolph
              yesterday


















            53












            $begingroup$

            The expression b + (c > 0 ? 1 : 2) is not a ternary operator; it is a function of two variables. There is one operation that results in $a$. You can certainly define a function
            $$f(b,c)=begin {cases} b+1&c gt 0\
            b+2 & c le 0 end {cases}$$



            You can also define functions with any number of inputs you want, so you can define $f(a,b,c)=a(b+c^2)$, for example. This is a ternary function.






            share|cite|improve this answer











            $endgroup$









            • 9




              $begingroup$
              Ternary operator in programming means a conditional statement of form "if A then x else y" and uually is written just as presented in the OP, i.e. A ? x : y. It would also literally be termary if the OP wouldn't have inserted the particular values of 1 and 2.
              $endgroup$
              – Džuris
              yesterday






            • 15




              $begingroup$
              @Džuris Ternary means "of arity 3", like binary means "of arity 2" and unary "of arity 1". The specific operator is the conditional ternary operator.
              $endgroup$
              – frabala
              yesterday






            • 15




              $begingroup$
              Once again, clarity beats compactness when it comes to mathematics.
              $endgroup$
              – Asaf Karagila
              yesterday






            • 6




              $begingroup$
              Well-chosen compact notations can contribute to clarity rather than detract from it, though.
              $endgroup$
              – Henning Makholm
              yesterday






            • 7




              $begingroup$
              @Džuris No it isn’t. The link you cite says, in its very first sentence, that “?: is a ternary operator” (not “the ternary operator”; emphasis mine). Some people refer to it as “the ternary operator” but this is strictly incorrect (and merits a correction e.g. on Stack Overflow).
              $endgroup$
              – Konrad Rudolph
              yesterday
















            53












            53








            53





            $begingroup$

            The expression b + (c > 0 ? 1 : 2) is not a ternary operator; it is a function of two variables. There is one operation that results in $a$. You can certainly define a function
            $$f(b,c)=begin {cases} b+1&c gt 0\
            b+2 & c le 0 end {cases}$$



            You can also define functions with any number of inputs you want, so you can define $f(a,b,c)=a(b+c^2)$, for example. This is a ternary function.






            share|cite|improve this answer











            $endgroup$



            The expression b + (c > 0 ? 1 : 2) is not a ternary operator; it is a function of two variables. There is one operation that results in $a$. You can certainly define a function
            $$f(b,c)=begin {cases} b+1&c gt 0\
            b+2 & c le 0 end {cases}$$



            You can also define functions with any number of inputs you want, so you can define $f(a,b,c)=a(b+c^2)$, for example. This is a ternary function.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 2 hours ago









            J. W. Tanner

            2,8211217




            2,8211217










            answered yesterday









            Ross MillikanRoss Millikan

            298k24200374




            298k24200374








            • 9




              $begingroup$
              Ternary operator in programming means a conditional statement of form "if A then x else y" and uually is written just as presented in the OP, i.e. A ? x : y. It would also literally be termary if the OP wouldn't have inserted the particular values of 1 and 2.
              $endgroup$
              – Džuris
              yesterday






            • 15




              $begingroup$
              @Džuris Ternary means "of arity 3", like binary means "of arity 2" and unary "of arity 1". The specific operator is the conditional ternary operator.
              $endgroup$
              – frabala
              yesterday






            • 15




              $begingroup$
              Once again, clarity beats compactness when it comes to mathematics.
              $endgroup$
              – Asaf Karagila
              yesterday






            • 6




              $begingroup$
              Well-chosen compact notations can contribute to clarity rather than detract from it, though.
              $endgroup$
              – Henning Makholm
              yesterday






            • 7




              $begingroup$
              @Džuris No it isn’t. The link you cite says, in its very first sentence, that “?: is a ternary operator” (not “the ternary operator”; emphasis mine). Some people refer to it as “the ternary operator” but this is strictly incorrect (and merits a correction e.g. on Stack Overflow).
              $endgroup$
              – Konrad Rudolph
              yesterday
















            • 9




              $begingroup$
              Ternary operator in programming means a conditional statement of form "if A then x else y" and uually is written just as presented in the OP, i.e. A ? x : y. It would also literally be termary if the OP wouldn't have inserted the particular values of 1 and 2.
              $endgroup$
              – Džuris
              yesterday






            • 15




              $begingroup$
              @Džuris Ternary means "of arity 3", like binary means "of arity 2" and unary "of arity 1". The specific operator is the conditional ternary operator.
              $endgroup$
              – frabala
              yesterday






            • 15




              $begingroup$
              Once again, clarity beats compactness when it comes to mathematics.
              $endgroup$
              – Asaf Karagila
              yesterday






            • 6




              $begingroup$
              Well-chosen compact notations can contribute to clarity rather than detract from it, though.
              $endgroup$
              – Henning Makholm
              yesterday






            • 7




              $begingroup$
              @Džuris No it isn’t. The link you cite says, in its very first sentence, that “?: is a ternary operator” (not “the ternary operator”; emphasis mine). Some people refer to it as “the ternary operator” but this is strictly incorrect (and merits a correction e.g. on Stack Overflow).
              $endgroup$
              – Konrad Rudolph
              yesterday










            9




            9




            $begingroup$
            Ternary operator in programming means a conditional statement of form "if A then x else y" and uually is written just as presented in the OP, i.e. A ? x : y. It would also literally be termary if the OP wouldn't have inserted the particular values of 1 and 2.
            $endgroup$
            – Džuris
            yesterday




            $begingroup$
            Ternary operator in programming means a conditional statement of form "if A then x else y" and uually is written just as presented in the OP, i.e. A ? x : y. It would also literally be termary if the OP wouldn't have inserted the particular values of 1 and 2.
            $endgroup$
            – Džuris
            yesterday




            15




            15




            $begingroup$
            @Džuris Ternary means "of arity 3", like binary means "of arity 2" and unary "of arity 1". The specific operator is the conditional ternary operator.
            $endgroup$
            – frabala
            yesterday




            $begingroup$
            @Džuris Ternary means "of arity 3", like binary means "of arity 2" and unary "of arity 1". The specific operator is the conditional ternary operator.
            $endgroup$
            – frabala
            yesterday




            15




            15




            $begingroup$
            Once again, clarity beats compactness when it comes to mathematics.
            $endgroup$
            – Asaf Karagila
            yesterday




            $begingroup$
            Once again, clarity beats compactness when it comes to mathematics.
            $endgroup$
            – Asaf Karagila
            yesterday




            6




            6




            $begingroup$
            Well-chosen compact notations can contribute to clarity rather than detract from it, though.
            $endgroup$
            – Henning Makholm
            yesterday




            $begingroup$
            Well-chosen compact notations can contribute to clarity rather than detract from it, though.
            $endgroup$
            – Henning Makholm
            yesterday




            7




            7




            $begingroup$
            @Džuris No it isn’t. The link you cite says, in its very first sentence, that “?: is a ternary operator” (not “the ternary operator”; emphasis mine). Some people refer to it as “the ternary operator” but this is strictly incorrect (and merits a correction e.g. on Stack Overflow).
            $endgroup$
            – Konrad Rudolph
            yesterday






            $begingroup$
            @Džuris No it isn’t. The link you cite says, in its very first sentence, that “?: is a ternary operator” (not “the ternary operator”; emphasis mine). Some people refer to it as “the ternary operator” but this is strictly incorrect (and merits a correction e.g. on Stack Overflow).
            $endgroup$
            – Konrad Rudolph
            yesterday













            35












            $begingroup$

            In Concrete Mathematics by Graham, Knuth and Patashnik, the authors use the "Iverson bracket" notation: Square brackets around a statement represent $1$ if the statement is true and $0$ otherwise. Using this notation, you could write
            $$
            a = b + 2 - [c gt 0].
            $$






            share|cite|improve this answer











            $endgroup$









            • 7




              $begingroup$
              aka en.wikipedia.org/wiki/Iverson_bracket
              $endgroup$
              – qwr
              yesterday










            • $begingroup$
              @qwr Thank you! I did not recall the name.
              $endgroup$
              – FredH
              yesterday






            • 5




              $begingroup$
              I would find it clearer to write $a = b + 1[c>0] + 2[cle0]$.
              $endgroup$
              – Rahul
              yesterday








            • 5




              $begingroup$
              I personally find Iverson brackets less cluttered than indicator functions, so I definitely recommend this notation.
              $endgroup$
              – J. M. is not a mathematician
              yesterday






            • 4




              $begingroup$
              Surely this is a unary operator, which works much like for example ! in the C programming language: Input is an arbitrary expression, the result is either 0 or 1 depending on the expression.
              $endgroup$
              – pipe
              yesterday


















            35












            $begingroup$

            In Concrete Mathematics by Graham, Knuth and Patashnik, the authors use the "Iverson bracket" notation: Square brackets around a statement represent $1$ if the statement is true and $0$ otherwise. Using this notation, you could write
            $$
            a = b + 2 - [c gt 0].
            $$






            share|cite|improve this answer











            $endgroup$









            • 7




              $begingroup$
              aka en.wikipedia.org/wiki/Iverson_bracket
              $endgroup$
              – qwr
              yesterday










            • $begingroup$
              @qwr Thank you! I did not recall the name.
              $endgroup$
              – FredH
              yesterday






            • 5




              $begingroup$
              I would find it clearer to write $a = b + 1[c>0] + 2[cle0]$.
              $endgroup$
              – Rahul
              yesterday








            • 5




              $begingroup$
              I personally find Iverson brackets less cluttered than indicator functions, so I definitely recommend this notation.
              $endgroup$
              – J. M. is not a mathematician
              yesterday






            • 4




              $begingroup$
              Surely this is a unary operator, which works much like for example ! in the C programming language: Input is an arbitrary expression, the result is either 0 or 1 depending on the expression.
              $endgroup$
              – pipe
              yesterday
















            35












            35








            35





            $begingroup$

            In Concrete Mathematics by Graham, Knuth and Patashnik, the authors use the "Iverson bracket" notation: Square brackets around a statement represent $1$ if the statement is true and $0$ otherwise. Using this notation, you could write
            $$
            a = b + 2 - [c gt 0].
            $$






            share|cite|improve this answer











            $endgroup$



            In Concrete Mathematics by Graham, Knuth and Patashnik, the authors use the "Iverson bracket" notation: Square brackets around a statement represent $1$ if the statement is true and $0$ otherwise. Using this notation, you could write
            $$
            a = b + 2 - [c gt 0].
            $$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited yesterday

























            answered yesterday









            FredHFredH

            839512




            839512








            • 7




              $begingroup$
              aka en.wikipedia.org/wiki/Iverson_bracket
              $endgroup$
              – qwr
              yesterday










            • $begingroup$
              @qwr Thank you! I did not recall the name.
              $endgroup$
              – FredH
              yesterday






            • 5




              $begingroup$
              I would find it clearer to write $a = b + 1[c>0] + 2[cle0]$.
              $endgroup$
              – Rahul
              yesterday








            • 5




              $begingroup$
              I personally find Iverson brackets less cluttered than indicator functions, so I definitely recommend this notation.
              $endgroup$
              – J. M. is not a mathematician
              yesterday






            • 4




              $begingroup$
              Surely this is a unary operator, which works much like for example ! in the C programming language: Input is an arbitrary expression, the result is either 0 or 1 depending on the expression.
              $endgroup$
              – pipe
              yesterday
















            • 7




              $begingroup$
              aka en.wikipedia.org/wiki/Iverson_bracket
              $endgroup$
              – qwr
              yesterday










            • $begingroup$
              @qwr Thank you! I did not recall the name.
              $endgroup$
              – FredH
              yesterday






            • 5




              $begingroup$
              I would find it clearer to write $a = b + 1[c>0] + 2[cle0]$.
              $endgroup$
              – Rahul
              yesterday








            • 5




              $begingroup$
              I personally find Iverson brackets less cluttered than indicator functions, so I definitely recommend this notation.
              $endgroup$
              – J. M. is not a mathematician
              yesterday






            • 4




              $begingroup$
              Surely this is a unary operator, which works much like for example ! in the C programming language: Input is an arbitrary expression, the result is either 0 or 1 depending on the expression.
              $endgroup$
              – pipe
              yesterday










            7




            7




            $begingroup$
            aka en.wikipedia.org/wiki/Iverson_bracket
            $endgroup$
            – qwr
            yesterday




            $begingroup$
            aka en.wikipedia.org/wiki/Iverson_bracket
            $endgroup$
            – qwr
            yesterday












            $begingroup$
            @qwr Thank you! I did not recall the name.
            $endgroup$
            – FredH
            yesterday




            $begingroup$
            @qwr Thank you! I did not recall the name.
            $endgroup$
            – FredH
            yesterday




            5




            5




            $begingroup$
            I would find it clearer to write $a = b + 1[c>0] + 2[cle0]$.
            $endgroup$
            – Rahul
            yesterday






            $begingroup$
            I would find it clearer to write $a = b + 1[c>0] + 2[cle0]$.
            $endgroup$
            – Rahul
            yesterday






            5




            5




            $begingroup$
            I personally find Iverson brackets less cluttered than indicator functions, so I definitely recommend this notation.
            $endgroup$
            – J. M. is not a mathematician
            yesterday




            $begingroup$
            I personally find Iverson brackets less cluttered than indicator functions, so I definitely recommend this notation.
            $endgroup$
            – J. M. is not a mathematician
            yesterday




            4




            4




            $begingroup$
            Surely this is a unary operator, which works much like for example ! in the C programming language: Input is an arbitrary expression, the result is either 0 or 1 depending on the expression.
            $endgroup$
            – pipe
            yesterday






            $begingroup$
            Surely this is a unary operator, which works much like for example ! in the C programming language: Input is an arbitrary expression, the result is either 0 or 1 depending on the expression.
            $endgroup$
            – pipe
            yesterday













            24












            $begingroup$

            Using the indicator function notation:$$a=b+1+mathbb{1}_{(-infty, 0]}(c)$$






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              Indicator is definitely the way to go, since the conditional can define an arbitrary set.
              $endgroup$
              – eyeballfrog
              yesterday










            • $begingroup$
              Thank you, that is very elegant. I will definitely try out all of the answers as I often use this in various scenarios.
              $endgroup$
              – dataphile
              yesterday






            • 2




              $begingroup$
              I often use - and have seen others use - $mathbb{1}$ ("blackboard bold") to denote the indicator function to distinguish it from the number $1$
              $endgroup$
              – Christopher
              yesterday








            • 1




              $begingroup$
              The indicator function isn’t a ternary operator, it only has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday


















            24












            $begingroup$

            Using the indicator function notation:$$a=b+1+mathbb{1}_{(-infty, 0]}(c)$$






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              Indicator is definitely the way to go, since the conditional can define an arbitrary set.
              $endgroup$
              – eyeballfrog
              yesterday










            • $begingroup$
              Thank you, that is very elegant. I will definitely try out all of the answers as I often use this in various scenarios.
              $endgroup$
              – dataphile
              yesterday






            • 2




              $begingroup$
              I often use - and have seen others use - $mathbb{1}$ ("blackboard bold") to denote the indicator function to distinguish it from the number $1$
              $endgroup$
              – Christopher
              yesterday








            • 1




              $begingroup$
              The indicator function isn’t a ternary operator, it only has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday
















            24












            24








            24





            $begingroup$

            Using the indicator function notation:$$a=b+1+mathbb{1}_{(-infty, 0]}(c)$$






            share|cite|improve this answer











            $endgroup$



            Using the indicator function notation:$$a=b+1+mathbb{1}_{(-infty, 0]}(c)$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited yesterday

























            answered yesterday









            Siong Thye GohSiong Thye Goh

            102k1467119




            102k1467119








            • 1




              $begingroup$
              Indicator is definitely the way to go, since the conditional can define an arbitrary set.
              $endgroup$
              – eyeballfrog
              yesterday










            • $begingroup$
              Thank you, that is very elegant. I will definitely try out all of the answers as I often use this in various scenarios.
              $endgroup$
              – dataphile
              yesterday






            • 2




              $begingroup$
              I often use - and have seen others use - $mathbb{1}$ ("blackboard bold") to denote the indicator function to distinguish it from the number $1$
              $endgroup$
              – Christopher
              yesterday








            • 1




              $begingroup$
              The indicator function isn’t a ternary operator, it only has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday
















            • 1




              $begingroup$
              Indicator is definitely the way to go, since the conditional can define an arbitrary set.
              $endgroup$
              – eyeballfrog
              yesterday










            • $begingroup$
              Thank you, that is very elegant. I will definitely try out all of the answers as I often use this in various scenarios.
              $endgroup$
              – dataphile
              yesterday






            • 2




              $begingroup$
              I often use - and have seen others use - $mathbb{1}$ ("blackboard bold") to denote the indicator function to distinguish it from the number $1$
              $endgroup$
              – Christopher
              yesterday








            • 1




              $begingroup$
              The indicator function isn’t a ternary operator, it only has two operands.
              $endgroup$
              – Konrad Rudolph
              yesterday










            1




            1




            $begingroup$
            Indicator is definitely the way to go, since the conditional can define an arbitrary set.
            $endgroup$
            – eyeballfrog
            yesterday




            $begingroup$
            Indicator is definitely the way to go, since the conditional can define an arbitrary set.
            $endgroup$
            – eyeballfrog
            yesterday












            $begingroup$
            Thank you, that is very elegant. I will definitely try out all of the answers as I often use this in various scenarios.
            $endgroup$
            – dataphile
            yesterday




            $begingroup$
            Thank you, that is very elegant. I will definitely try out all of the answers as I often use this in various scenarios.
            $endgroup$
            – dataphile
            yesterday




            2




            2




            $begingroup$
            I often use - and have seen others use - $mathbb{1}$ ("blackboard bold") to denote the indicator function to distinguish it from the number $1$
            $endgroup$
            – Christopher
            yesterday






            $begingroup$
            I often use - and have seen others use - $mathbb{1}$ ("blackboard bold") to denote the indicator function to distinguish it from the number $1$
            $endgroup$
            – Christopher
            yesterday






            1




            1




            $begingroup$
            The indicator function isn’t a ternary operator, it only has two operands.
            $endgroup$
            – Konrad Rudolph
            yesterday






            $begingroup$
            The indicator function isn’t a ternary operator, it only has two operands.
            $endgroup$
            – Konrad Rudolph
            yesterday













            13












            $begingroup$

            In math, equations are written in piecewise form by having a curly brace enclose multiple lines; each one with a condition excepting the last which has "otherwise".



            There are a few custom operators that also occasionally make an appearance. E.g. the Heavyside function mentioned by Alex, the Dirac function, and the cyclical operator $delta_{ijk}$ - all of which can be used to emulate conditional behaviour.






            share|cite|improve this answer









            $endgroup$


















              13












              $begingroup$

              In math, equations are written in piecewise form by having a curly brace enclose multiple lines; each one with a condition excepting the last which has "otherwise".



              There are a few custom operators that also occasionally make an appearance. E.g. the Heavyside function mentioned by Alex, the Dirac function, and the cyclical operator $delta_{ijk}$ - all of which can be used to emulate conditional behaviour.






              share|cite|improve this answer









              $endgroup$
















                13












                13








                13





                $begingroup$

                In math, equations are written in piecewise form by having a curly brace enclose multiple lines; each one with a condition excepting the last which has "otherwise".



                There are a few custom operators that also occasionally make an appearance. E.g. the Heavyside function mentioned by Alex, the Dirac function, and the cyclical operator $delta_{ijk}$ - all of which can be used to emulate conditional behaviour.






                share|cite|improve this answer









                $endgroup$



                In math, equations are written in piecewise form by having a curly brace enclose multiple lines; each one with a condition excepting the last which has "otherwise".



                There are a few custom operators that also occasionally make an appearance. E.g. the Heavyside function mentioned by Alex, the Dirac function, and the cyclical operator $delta_{ijk}$ - all of which can be used to emulate conditional behaviour.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered yesterday









                Paul ChildsPaul Childs

                4138




                4138























                    4












                    $begingroup$

                    One should realize that operators are just a fancy way of using functions.
                    So a ternary operator is a function of 3 variables that is notated in a different way. Is that useful? The answer is mostly not. Also realize that any mathematician is allowed to introduce any notation he feels is illustrative.



                    Let's review why we use binary operators at all like in a+b*c . Because parameters and results are of the same type, it makes sense to leave out parentheses and introduce complicated priority rules. Imagine that a b c are numbers and we have a normal + and a peculiar * that results in dragons. Now the expression doesn't make sense (assumming a high priority *), because there is no way to add numbers and dragons. Thusly most ternary operators results in a mess.



                    With a proper notation there are examples of ternary operations. For example, there is a special notation for "sum for i from a to b of expression". This takes two boundaries (numbers) and a function from a number of that type that results in another number. (Mathematician, read "element of an addition group" for number.)
                    The notation for integration is similarly ternary.

                    So in short ternary operators exist, and you can define your own. They are in general accompagnied with a special notation, or they are not helpful.



                    Now back to the special case you mention.
                    Because truth values are implied in math, an expression like "if a then b else c" makes sense if a represens a truth value like (7<12). The above expression is understood in every mathematical context. However in a context where truth values are not considered a set, (if .. then .. else ..) would not be considered an operator/function, but a textual explanation. A general accepted notation could be useful in math, but I'm not aware there is one. That is probably, because like in the above, informal notations are readily understood.






                    share|cite|improve this answer









                    $endgroup$


















                      4












                      $begingroup$

                      One should realize that operators are just a fancy way of using functions.
                      So a ternary operator is a function of 3 variables that is notated in a different way. Is that useful? The answer is mostly not. Also realize that any mathematician is allowed to introduce any notation he feels is illustrative.



                      Let's review why we use binary operators at all like in a+b*c . Because parameters and results are of the same type, it makes sense to leave out parentheses and introduce complicated priority rules. Imagine that a b c are numbers and we have a normal + and a peculiar * that results in dragons. Now the expression doesn't make sense (assumming a high priority *), because there is no way to add numbers and dragons. Thusly most ternary operators results in a mess.



                      With a proper notation there are examples of ternary operations. For example, there is a special notation for "sum for i from a to b of expression". This takes two boundaries (numbers) and a function from a number of that type that results in another number. (Mathematician, read "element of an addition group" for number.)
                      The notation for integration is similarly ternary.

                      So in short ternary operators exist, and you can define your own. They are in general accompagnied with a special notation, or they are not helpful.



                      Now back to the special case you mention.
                      Because truth values are implied in math, an expression like "if a then b else c" makes sense if a represens a truth value like (7<12). The above expression is understood in every mathematical context. However in a context where truth values are not considered a set, (if .. then .. else ..) would not be considered an operator/function, but a textual explanation. A general accepted notation could be useful in math, but I'm not aware there is one. That is probably, because like in the above, informal notations are readily understood.






                      share|cite|improve this answer









                      $endgroup$
















                        4












                        4








                        4





                        $begingroup$

                        One should realize that operators are just a fancy way of using functions.
                        So a ternary operator is a function of 3 variables that is notated in a different way. Is that useful? The answer is mostly not. Also realize that any mathematician is allowed to introduce any notation he feels is illustrative.



                        Let's review why we use binary operators at all like in a+b*c . Because parameters and results are of the same type, it makes sense to leave out parentheses and introduce complicated priority rules. Imagine that a b c are numbers and we have a normal + and a peculiar * that results in dragons. Now the expression doesn't make sense (assumming a high priority *), because there is no way to add numbers and dragons. Thusly most ternary operators results in a mess.



                        With a proper notation there are examples of ternary operations. For example, there is a special notation for "sum for i from a to b of expression". This takes two boundaries (numbers) and a function from a number of that type that results in another number. (Mathematician, read "element of an addition group" for number.)
                        The notation for integration is similarly ternary.

                        So in short ternary operators exist, and you can define your own. They are in general accompagnied with a special notation, or they are not helpful.



                        Now back to the special case you mention.
                        Because truth values are implied in math, an expression like "if a then b else c" makes sense if a represens a truth value like (7<12). The above expression is understood in every mathematical context. However in a context where truth values are not considered a set, (if .. then .. else ..) would not be considered an operator/function, but a textual explanation. A general accepted notation could be useful in math, but I'm not aware there is one. That is probably, because like in the above, informal notations are readily understood.






                        share|cite|improve this answer









                        $endgroup$



                        One should realize that operators are just a fancy way of using functions.
                        So a ternary operator is a function of 3 variables that is notated in a different way. Is that useful? The answer is mostly not. Also realize that any mathematician is allowed to introduce any notation he feels is illustrative.



                        Let's review why we use binary operators at all like in a+b*c . Because parameters and results are of the same type, it makes sense to leave out parentheses and introduce complicated priority rules. Imagine that a b c are numbers and we have a normal + and a peculiar * that results in dragons. Now the expression doesn't make sense (assumming a high priority *), because there is no way to add numbers and dragons. Thusly most ternary operators results in a mess.



                        With a proper notation there are examples of ternary operations. For example, there is a special notation for "sum for i from a to b of expression". This takes two boundaries (numbers) and a function from a number of that type that results in another number. (Mathematician, read "element of an addition group" for number.)
                        The notation for integration is similarly ternary.

                        So in short ternary operators exist, and you can define your own. They are in general accompagnied with a special notation, or they are not helpful.



                        Now back to the special case you mention.
                        Because truth values are implied in math, an expression like "if a then b else c" makes sense if a represens a truth value like (7<12). The above expression is understood in every mathematical context. However in a context where truth values are not considered a set, (if .. then .. else ..) would not be considered an operator/function, but a textual explanation. A general accepted notation could be useful in math, but I'm not aware there is one. That is probably, because like in the above, informal notations are readily understood.







                        share|cite|improve this answer












                        share|cite|improve this answer



                        share|cite|improve this answer










                        answered yesterday









                        Albert van der HorstAlbert van der Horst

                        19115




                        19115























                            2












                            $begingroup$

                            Following solution is not defined for $c = 0$; however it uses very basic operations only, which might be useful as you probably look for an expression to implement in a program:



                            $$a = b + 1lambda + 2(1-lambda)$$



                            where



                            $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$





                            You need to make the problem discrete and make a choice from two values. So, given some value $c in mathbb{R}$ we need to calculate some value $lambda in {0,1}$ depnding on $c<0$ or $c>0$.



                            Knowing that



                            $$frac{|c|}{c} in {1,-1}$$



                            we can calculate the $lambda$ as follows:



                            $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$



                            Now that our $lambda in {0,1}$ we can do the "choice" between the two constants $d$ and $e$ as follows:



                            $$dlambda + e(1-lambda)$$



                            which equals $d$ for $lambda = 1$, and $e$ for $lambda = 0$.






                            share|cite|improve this answer










                            New contributor




                            Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                            Check out our Code of Conduct.






                            $endgroup$


















                              2












                              $begingroup$

                              Following solution is not defined for $c = 0$; however it uses very basic operations only, which might be useful as you probably look for an expression to implement in a program:



                              $$a = b + 1lambda + 2(1-lambda)$$



                              where



                              $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$





                              You need to make the problem discrete and make a choice from two values. So, given some value $c in mathbb{R}$ we need to calculate some value $lambda in {0,1}$ depnding on $c<0$ or $c>0$.



                              Knowing that



                              $$frac{|c|}{c} in {1,-1}$$



                              we can calculate the $lambda$ as follows:



                              $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$



                              Now that our $lambda in {0,1}$ we can do the "choice" between the two constants $d$ and $e$ as follows:



                              $$dlambda + e(1-lambda)$$



                              which equals $d$ for $lambda = 1$, and $e$ for $lambda = 0$.






                              share|cite|improve this answer










                              New contributor




                              Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                              Check out our Code of Conduct.






                              $endgroup$
















                                2












                                2








                                2





                                $begingroup$

                                Following solution is not defined for $c = 0$; however it uses very basic operations only, which might be useful as you probably look for an expression to implement in a program:



                                $$a = b + 1lambda + 2(1-lambda)$$



                                where



                                $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$





                                You need to make the problem discrete and make a choice from two values. So, given some value $c in mathbb{R}$ we need to calculate some value $lambda in {0,1}$ depnding on $c<0$ or $c>0$.



                                Knowing that



                                $$frac{|c|}{c} in {1,-1}$$



                                we can calculate the $lambda$ as follows:



                                $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$



                                Now that our $lambda in {0,1}$ we can do the "choice" between the two constants $d$ and $e$ as follows:



                                $$dlambda + e(1-lambda)$$



                                which equals $d$ for $lambda = 1$, and $e$ for $lambda = 0$.






                                share|cite|improve this answer










                                New contributor




                                Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                Check out our Code of Conduct.






                                $endgroup$



                                Following solution is not defined for $c = 0$; however it uses very basic operations only, which might be useful as you probably look for an expression to implement in a program:



                                $$a = b + 1lambda + 2(1-lambda)$$



                                where



                                $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$





                                You need to make the problem discrete and make a choice from two values. So, given some value $c in mathbb{R}$ we need to calculate some value $lambda in {0,1}$ depnding on $c<0$ or $c>0$.



                                Knowing that



                                $$frac{|c|}{c} in {1,-1}$$



                                we can calculate the $lambda$ as follows:



                                $$lambda = frac{ 1 + frac{|c|}{c} }{2}$$



                                Now that our $lambda in {0,1}$ we can do the "choice" between the two constants $d$ and $e$ as follows:



                                $$dlambda + e(1-lambda)$$



                                which equals $d$ for $lambda = 1$, and $e$ for $lambda = 0$.







                                share|cite|improve this answer










                                New contributor




                                Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                Check out our Code of Conduct.









                                share|cite|improve this answer



                                share|cite|improve this answer








                                edited 12 hours ago





















                                New contributor




                                Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                Check out our Code of Conduct.









                                answered 12 hours ago









                                TimurTimur

                                1213




                                1213




                                New contributor




                                Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                Check out our Code of Conduct.





                                New contributor





                                Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                Check out our Code of Conduct.






                                Timur is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                                Check out our Code of Conduct.























                                    1












                                    $begingroup$

                                    There are many good answers that give notation for, “if this condition holds, then 1, else 0.” This corresponds to an even simpler expression in C;(x>1) is equivalent to (x>1 ? 1 : 0).



                                    It’s worth noting that the ternary operator is more general than that. If the arguments are elements of a ring, you could express c ? a : b with (using Iverson-bracket notation) $(a-b) cdot [c] + b$, but not otherwise. (And compilers frequently use this trick, in a Boolean ring, to compile conditionals without needing to execute a branch instruction.) In a C program, evaluating the expressions $a$ or $b$ might have side-effects, such as deleting a file or printing a message to the screen. In a mathematical function, this isn’t something you would worry about, and a programming language where this is impossible is called functional.



                                    Ross Millikan gave the most standard notation, a cases block. The closest equivalent in mathematical computer science is the if-then-else function of Lambda Calculus.






                                    share|cite|improve this answer











                                    $endgroup$













                                    • $begingroup$
                                      "in C;(x>1) is equivalent to (x>1 ? 1 : 0)" Not exactly. Rather, C considers any nonzero value as equivalent from a truthiness perspective. So there is no difference between an integer expression taking the value of -1, 1, 42 or INT_MAX when that expression is treated as a boolean rvalue. In C, the one special integer value, when treated as a boolean, is 0, representing false. That said, if someone actually used (x>1) as a non-boolean expression and I noticed it, I would likely at least briefly try to find some heavy physical object that could be applied at high speed to their keyboard.
                                      $endgroup$
                                      – a CVn
                                      yesterday






                                    • 1




                                      $begingroup$
                                      @aCVn The C11 standard states, “Each of the operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) shall yield 1 if the specified relation is true and 0 if it is false.” Similarly for equality and inequality, “Each of the operators yields 1 if the specified relation is true and 0 if it is false.”
                                      $endgroup$
                                      – Davislor
                                      yesterday












                                    • $begingroup$
                                      @aCVn What you wrote is true, but the relational operators in C will only return the values 1 or 0.
                                      $endgroup$
                                      – Davislor
                                      yesterday
















                                    1












                                    $begingroup$

                                    There are many good answers that give notation for, “if this condition holds, then 1, else 0.” This corresponds to an even simpler expression in C;(x>1) is equivalent to (x>1 ? 1 : 0).



                                    It’s worth noting that the ternary operator is more general than that. If the arguments are elements of a ring, you could express c ? a : b with (using Iverson-bracket notation) $(a-b) cdot [c] + b$, but not otherwise. (And compilers frequently use this trick, in a Boolean ring, to compile conditionals without needing to execute a branch instruction.) In a C program, evaluating the expressions $a$ or $b$ might have side-effects, such as deleting a file or printing a message to the screen. In a mathematical function, this isn’t something you would worry about, and a programming language where this is impossible is called functional.



                                    Ross Millikan gave the most standard notation, a cases block. The closest equivalent in mathematical computer science is the if-then-else function of Lambda Calculus.






                                    share|cite|improve this answer











                                    $endgroup$













                                    • $begingroup$
                                      "in C;(x>1) is equivalent to (x>1 ? 1 : 0)" Not exactly. Rather, C considers any nonzero value as equivalent from a truthiness perspective. So there is no difference between an integer expression taking the value of -1, 1, 42 or INT_MAX when that expression is treated as a boolean rvalue. In C, the one special integer value, when treated as a boolean, is 0, representing false. That said, if someone actually used (x>1) as a non-boolean expression and I noticed it, I would likely at least briefly try to find some heavy physical object that could be applied at high speed to their keyboard.
                                      $endgroup$
                                      – a CVn
                                      yesterday






                                    • 1




                                      $begingroup$
                                      @aCVn The C11 standard states, “Each of the operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) shall yield 1 if the specified relation is true and 0 if it is false.” Similarly for equality and inequality, “Each of the operators yields 1 if the specified relation is true and 0 if it is false.”
                                      $endgroup$
                                      – Davislor
                                      yesterday












                                    • $begingroup$
                                      @aCVn What you wrote is true, but the relational operators in C will only return the values 1 or 0.
                                      $endgroup$
                                      – Davislor
                                      yesterday














                                    1












                                    1








                                    1





                                    $begingroup$

                                    There are many good answers that give notation for, “if this condition holds, then 1, else 0.” This corresponds to an even simpler expression in C;(x>1) is equivalent to (x>1 ? 1 : 0).



                                    It’s worth noting that the ternary operator is more general than that. If the arguments are elements of a ring, you could express c ? a : b with (using Iverson-bracket notation) $(a-b) cdot [c] + b$, but not otherwise. (And compilers frequently use this trick, in a Boolean ring, to compile conditionals without needing to execute a branch instruction.) In a C program, evaluating the expressions $a$ or $b$ might have side-effects, such as deleting a file or printing a message to the screen. In a mathematical function, this isn’t something you would worry about, and a programming language where this is impossible is called functional.



                                    Ross Millikan gave the most standard notation, a cases block. The closest equivalent in mathematical computer science is the if-then-else function of Lambda Calculus.






                                    share|cite|improve this answer











                                    $endgroup$



                                    There are many good answers that give notation for, “if this condition holds, then 1, else 0.” This corresponds to an even simpler expression in C;(x>1) is equivalent to (x>1 ? 1 : 0).



                                    It’s worth noting that the ternary operator is more general than that. If the arguments are elements of a ring, you could express c ? a : b with (using Iverson-bracket notation) $(a-b) cdot [c] + b$, but not otherwise. (And compilers frequently use this trick, in a Boolean ring, to compile conditionals without needing to execute a branch instruction.) In a C program, evaluating the expressions $a$ or $b$ might have side-effects, such as deleting a file or printing a message to the screen. In a mathematical function, this isn’t something you would worry about, and a programming language where this is impossible is called functional.



                                    Ross Millikan gave the most standard notation, a cases block. The closest equivalent in mathematical computer science is the if-then-else function of Lambda Calculus.







                                    share|cite|improve this answer














                                    share|cite|improve this answer



                                    share|cite|improve this answer








                                    edited yesterday

























                                    answered yesterday









                                    DavislorDavislor

                                    2,360815




                                    2,360815












                                    • $begingroup$
                                      "in C;(x>1) is equivalent to (x>1 ? 1 : 0)" Not exactly. Rather, C considers any nonzero value as equivalent from a truthiness perspective. So there is no difference between an integer expression taking the value of -1, 1, 42 or INT_MAX when that expression is treated as a boolean rvalue. In C, the one special integer value, when treated as a boolean, is 0, representing false. That said, if someone actually used (x>1) as a non-boolean expression and I noticed it, I would likely at least briefly try to find some heavy physical object that could be applied at high speed to their keyboard.
                                      $endgroup$
                                      – a CVn
                                      yesterday






                                    • 1




                                      $begingroup$
                                      @aCVn The C11 standard states, “Each of the operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) shall yield 1 if the specified relation is true and 0 if it is false.” Similarly for equality and inequality, “Each of the operators yields 1 if the specified relation is true and 0 if it is false.”
                                      $endgroup$
                                      – Davislor
                                      yesterday












                                    • $begingroup$
                                      @aCVn What you wrote is true, but the relational operators in C will only return the values 1 or 0.
                                      $endgroup$
                                      – Davislor
                                      yesterday


















                                    • $begingroup$
                                      "in C;(x>1) is equivalent to (x>1 ? 1 : 0)" Not exactly. Rather, C considers any nonzero value as equivalent from a truthiness perspective. So there is no difference between an integer expression taking the value of -1, 1, 42 or INT_MAX when that expression is treated as a boolean rvalue. In C, the one special integer value, when treated as a boolean, is 0, representing false. That said, if someone actually used (x>1) as a non-boolean expression and I noticed it, I would likely at least briefly try to find some heavy physical object that could be applied at high speed to their keyboard.
                                      $endgroup$
                                      – a CVn
                                      yesterday






                                    • 1




                                      $begingroup$
                                      @aCVn The C11 standard states, “Each of the operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) shall yield 1 if the specified relation is true and 0 if it is false.” Similarly for equality and inequality, “Each of the operators yields 1 if the specified relation is true and 0 if it is false.”
                                      $endgroup$
                                      – Davislor
                                      yesterday












                                    • $begingroup$
                                      @aCVn What you wrote is true, but the relational operators in C will only return the values 1 or 0.
                                      $endgroup$
                                      – Davislor
                                      yesterday
















                                    $begingroup$
                                    "in C;(x>1) is equivalent to (x>1 ? 1 : 0)" Not exactly. Rather, C considers any nonzero value as equivalent from a truthiness perspective. So there is no difference between an integer expression taking the value of -1, 1, 42 or INT_MAX when that expression is treated as a boolean rvalue. In C, the one special integer value, when treated as a boolean, is 0, representing false. That said, if someone actually used (x>1) as a non-boolean expression and I noticed it, I would likely at least briefly try to find some heavy physical object that could be applied at high speed to their keyboard.
                                    $endgroup$
                                    – a CVn
                                    yesterday




                                    $begingroup$
                                    "in C;(x>1) is equivalent to (x>1 ? 1 : 0)" Not exactly. Rather, C considers any nonzero value as equivalent from a truthiness perspective. So there is no difference between an integer expression taking the value of -1, 1, 42 or INT_MAX when that expression is treated as a boolean rvalue. In C, the one special integer value, when treated as a boolean, is 0, representing false. That said, if someone actually used (x>1) as a non-boolean expression and I noticed it, I would likely at least briefly try to find some heavy physical object that could be applied at high speed to their keyboard.
                                    $endgroup$
                                    – a CVn
                                    yesterday




                                    1




                                    1




                                    $begingroup$
                                    @aCVn The C11 standard states, “Each of the operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) shall yield 1 if the specified relation is true and 0 if it is false.” Similarly for equality and inequality, “Each of the operators yields 1 if the specified relation is true and 0 if it is false.”
                                    $endgroup$
                                    – Davislor
                                    yesterday






                                    $begingroup$
                                    @aCVn The C11 standard states, “Each of the operators < (less than), > (greater than), <= (less than or equal to), and >= (greater than or equal to) shall yield 1 if the specified relation is true and 0 if it is false.” Similarly for equality and inequality, “Each of the operators yields 1 if the specified relation is true and 0 if it is false.”
                                    $endgroup$
                                    – Davislor
                                    yesterday














                                    $begingroup$
                                    @aCVn What you wrote is true, but the relational operators in C will only return the values 1 or 0.
                                    $endgroup$
                                    – Davislor
                                    yesterday




                                    $begingroup$
                                    @aCVn What you wrote is true, but the relational operators in C will only return the values 1 or 0.
                                    $endgroup$
                                    – Davislor
                                    yesterday











                                    1












                                    $begingroup$

                                    I think mathematicians should not be afraid to use the Iverson bracket, including when teaching, as this is a generally very useful notation whose only slightly unusual feature is to introduce a logical expression in the middle of an algebraic one (but one already regularly finds conditions inside set-theoretic expressions, so it really is not a big deal). It may avoid a lot of clutter, notably many instances of clumsy expressions by cases with a big unmatched brace (which is usually only usable as the right hand side of a definition). Since brackets do have many other uses in mathematics, I personally prefer a typographically distinct representation of Iverson brackets, rendering your example as
                                    $$def[#1]{[![{#1}]!]}
                                    a= b + [c>0]1 + [cnot>0]2.
                                    $$

                                    This works best in additive context (though one can use Iverson brackets in the exponent for optional multiplicative factors). It is not really ideal for general two-way branches, as the condition must be repeated twice, one of them in negated form, but it happens that most of the time one needs $0$ as the value for one branch anyway.



                                    As a more concise two-way branch, I can recall that Algol68 introduced the notation $b+(c>0mid 1mid 2)$ for the right-hand side of your equation; though this is a programming language and not mathematics, it was designed by mathematicians. They also had notation for multi-way branching: thus the solution to the recursion $a_{n+2}=a_{n+1}-a_n$ with initial values $a_0=0$, $a_1=1$ can be written
                                    $$
                                    a_n=(nbmod 6+1mid 0,1,1,0,-1,-1)
                                    $$

                                    (where the "${}+1$" is needed because in 1968 they still counted starting from $1$, which is a mistake), which is reasonably concise and readable, compared to other ways to express this result. Also consider, for month $m$ in year $y$, the number
                                    $$
                                    ( m mid 31,(ybmod 4=0land ybmod 100neq0lor ybmod400=0mid 29mid 28)
                                    ,31,30,31,30,31,31,30,31,30,31).
                                    $$






                                    share|cite|improve this answer









                                    $endgroup$


















                                      1












                                      $begingroup$

                                      I think mathematicians should not be afraid to use the Iverson bracket, including when teaching, as this is a generally very useful notation whose only slightly unusual feature is to introduce a logical expression in the middle of an algebraic one (but one already regularly finds conditions inside set-theoretic expressions, so it really is not a big deal). It may avoid a lot of clutter, notably many instances of clumsy expressions by cases with a big unmatched brace (which is usually only usable as the right hand side of a definition). Since brackets do have many other uses in mathematics, I personally prefer a typographically distinct representation of Iverson brackets, rendering your example as
                                      $$def[#1]{[![{#1}]!]}
                                      a= b + [c>0]1 + [cnot>0]2.
                                      $$

                                      This works best in additive context (though one can use Iverson brackets in the exponent for optional multiplicative factors). It is not really ideal for general two-way branches, as the condition must be repeated twice, one of them in negated form, but it happens that most of the time one needs $0$ as the value for one branch anyway.



                                      As a more concise two-way branch, I can recall that Algol68 introduced the notation $b+(c>0mid 1mid 2)$ for the right-hand side of your equation; though this is a programming language and not mathematics, it was designed by mathematicians. They also had notation for multi-way branching: thus the solution to the recursion $a_{n+2}=a_{n+1}-a_n$ with initial values $a_0=0$, $a_1=1$ can be written
                                      $$
                                      a_n=(nbmod 6+1mid 0,1,1,0,-1,-1)
                                      $$

                                      (where the "${}+1$" is needed because in 1968 they still counted starting from $1$, which is a mistake), which is reasonably concise and readable, compared to other ways to express this result. Also consider, for month $m$ in year $y$, the number
                                      $$
                                      ( m mid 31,(ybmod 4=0land ybmod 100neq0lor ybmod400=0mid 29mid 28)
                                      ,31,30,31,30,31,31,30,31,30,31).
                                      $$






                                      share|cite|improve this answer









                                      $endgroup$
















                                        1












                                        1








                                        1





                                        $begingroup$

                                        I think mathematicians should not be afraid to use the Iverson bracket, including when teaching, as this is a generally very useful notation whose only slightly unusual feature is to introduce a logical expression in the middle of an algebraic one (but one already regularly finds conditions inside set-theoretic expressions, so it really is not a big deal). It may avoid a lot of clutter, notably many instances of clumsy expressions by cases with a big unmatched brace (which is usually only usable as the right hand side of a definition). Since brackets do have many other uses in mathematics, I personally prefer a typographically distinct representation of Iverson brackets, rendering your example as
                                        $$def[#1]{[![{#1}]!]}
                                        a= b + [c>0]1 + [cnot>0]2.
                                        $$

                                        This works best in additive context (though one can use Iverson brackets in the exponent for optional multiplicative factors). It is not really ideal for general two-way branches, as the condition must be repeated twice, one of them in negated form, but it happens that most of the time one needs $0$ as the value for one branch anyway.



                                        As a more concise two-way branch, I can recall that Algol68 introduced the notation $b+(c>0mid 1mid 2)$ for the right-hand side of your equation; though this is a programming language and not mathematics, it was designed by mathematicians. They also had notation for multi-way branching: thus the solution to the recursion $a_{n+2}=a_{n+1}-a_n$ with initial values $a_0=0$, $a_1=1$ can be written
                                        $$
                                        a_n=(nbmod 6+1mid 0,1,1,0,-1,-1)
                                        $$

                                        (where the "${}+1$" is needed because in 1968 they still counted starting from $1$, which is a mistake), which is reasonably concise and readable, compared to other ways to express this result. Also consider, for month $m$ in year $y$, the number
                                        $$
                                        ( m mid 31,(ybmod 4=0land ybmod 100neq0lor ybmod400=0mid 29mid 28)
                                        ,31,30,31,30,31,31,30,31,30,31).
                                        $$






                                        share|cite|improve this answer









                                        $endgroup$



                                        I think mathematicians should not be afraid to use the Iverson bracket, including when teaching, as this is a generally very useful notation whose only slightly unusual feature is to introduce a logical expression in the middle of an algebraic one (but one already regularly finds conditions inside set-theoretic expressions, so it really is not a big deal). It may avoid a lot of clutter, notably many instances of clumsy expressions by cases with a big unmatched brace (which is usually only usable as the right hand side of a definition). Since brackets do have many other uses in mathematics, I personally prefer a typographically distinct representation of Iverson brackets, rendering your example as
                                        $$def[#1]{[![{#1}]!]}
                                        a= b + [c>0]1 + [cnot>0]2.
                                        $$

                                        This works best in additive context (though one can use Iverson brackets in the exponent for optional multiplicative factors). It is not really ideal for general two-way branches, as the condition must be repeated twice, one of them in negated form, but it happens that most of the time one needs $0$ as the value for one branch anyway.



                                        As a more concise two-way branch, I can recall that Algol68 introduced the notation $b+(c>0mid 1mid 2)$ for the right-hand side of your equation; though this is a programming language and not mathematics, it was designed by mathematicians. They also had notation for multi-way branching: thus the solution to the recursion $a_{n+2}=a_{n+1}-a_n$ with initial values $a_0=0$, $a_1=1$ can be written
                                        $$
                                        a_n=(nbmod 6+1mid 0,1,1,0,-1,-1)
                                        $$

                                        (where the "${}+1$" is needed because in 1968 they still counted starting from $1$, which is a mistake), which is reasonably concise and readable, compared to other ways to express this result. Also consider, for month $m$ in year $y$, the number
                                        $$
                                        ( m mid 31,(ybmod 4=0land ybmod 100neq0lor ybmod400=0mid 29mid 28)
                                        ,31,30,31,30,31,31,30,31,30,31).
                                        $$







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                                        answered 12 hours ago









                                        Marc van LeeuwenMarc van Leeuwen

                                        87.8k5111225




                                        87.8k5111225























                                            0












                                            $begingroup$

                                            It can be viewed as inline function:



                                            $$f(c)=begin {cases} 1&c gt 0\
                                            2& c le 0 end {cases}$$



                                            so it comes as:



                                            $$a = b + f(c)$$
                                            used in everyday mathematics.






                                            share|cite|improve this answer









                                            $endgroup$


















                                              0












                                              $begingroup$

                                              It can be viewed as inline function:



                                              $$f(c)=begin {cases} 1&c gt 0\
                                              2& c le 0 end {cases}$$



                                              so it comes as:



                                              $$a = b + f(c)$$
                                              used in everyday mathematics.






                                              share|cite|improve this answer









                                              $endgroup$
















                                                0












                                                0








                                                0





                                                $begingroup$

                                                It can be viewed as inline function:



                                                $$f(c)=begin {cases} 1&c gt 0\
                                                2& c le 0 end {cases}$$



                                                so it comes as:



                                                $$a = b + f(c)$$
                                                used in everyday mathematics.






                                                share|cite|improve this answer









                                                $endgroup$



                                                It can be viewed as inline function:



                                                $$f(c)=begin {cases} 1&c gt 0\
                                                2& c le 0 end {cases}$$



                                                so it comes as:



                                                $$a = b + f(c)$$
                                                used in everyday mathematics.







                                                share|cite|improve this answer












                                                share|cite|improve this answer



                                                share|cite|improve this answer










                                                answered 21 hours ago









                                                yekanchiyekanchi

                                                1012




                                                1012






























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