Why do we say 'Pairwise Disjoint', rather than 'Disjoint'?If $A$ is infinite then it has two infinite subsets...
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Why do we say 'Pairwise Disjoint', rather than 'Disjoint'?
If $A$ is infinite then it has two infinite subsets $B, C$ which are pairwise disjoint.Pairwise disjoint proofDistinction between the notions of pairwise disjointPairwise and Mutually disjoint setsIs the empty family of sets pairwise disjoint?Do Kolmogorov's axioms really need only disjointness rather than pairwise disjointness?Proof containing pairwise disjoint setsA set whose power set is pairwise disjoint?pairwise disjoint , or disjointProve that sets are pairwise disjoint
$begingroup$
I don't see the ambiguity that 'Pairwise' resolves.
Surely if A,B,C are disjoint sets then they are pairwise disjoint and vice versa?
Or am I being dim?
elementary-set-theory
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
$endgroup$
add a comment |
$begingroup$
I don't see the ambiguity that 'Pairwise' resolves.
Surely if A,B,C are disjoint sets then they are pairwise disjoint and vice versa?
Or am I being dim?
elementary-set-theory
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
$endgroup$
add a comment |
$begingroup$
I don't see the ambiguity that 'Pairwise' resolves.
Surely if A,B,C are disjoint sets then they are pairwise disjoint and vice versa?
Or am I being dim?
elementary-set-theory
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
$endgroup$
I don't see the ambiguity that 'Pairwise' resolves.
Surely if A,B,C are disjoint sets then they are pairwise disjoint and vice versa?
Or am I being dim?
elementary-set-theory
elementary-set-theory
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
asked 1 hour ago
John Lawrence AspdenJohn Lawrence Aspden
25517
25517
add a comment |
add a comment |
5 Answers
5
active
oldest
votes
$begingroup$
${1,2},{2,3},{1,3}$ are disjoint but not pairwise disjoint.
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
$endgroup$
$begingroup$
Really? Who would call those disjoint sets?
$endgroup$
– John Lawrence Aspden
56 mins ago
1
$begingroup$
Everyone. Disjoint means their intersection is empty.
$endgroup$
– saulspatz
55 mins ago
$begingroup$
If that's true then I'll accept the answer (and thanks!). Can you cite or give a popular textbook that uses this definition?
$endgroup$
– John Lawrence Aspden
53 mins ago
$begingroup$
Sorry, I don't have any elementary textbooks any more.
$endgroup$
– saulspatz
50 mins ago
2
$begingroup$
If sets $A_1,A_2,dots,A_n$ are said to be disjoint then usually it is meant that the sets are pairwise disjoint. See Wolfram for instance. In the other case one says simply that the sets have an empty intersection.
$endgroup$
– drhab
32 mins ago
|
show 3 more comments
$begingroup$
In this context disjoint means $A cap B cap C = emptyset$.
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
$endgroup$
3
$begingroup$
Is that a standard meaning? I never saw the term formally defined that way in four years of undergrad math classes.
$endgroup$
– Connor Harris
56 mins ago
$begingroup$
me neither, but four people have answered the question this way in four minutes!
$endgroup$
– John Lawrence Aspden
51 mins ago
$begingroup$
If you define "disjoint" to mean "empty intersection" (which is the standard definition) then formally for a family of sets "disjoint" would mean the intersection of the entire family is empty unless stated otherwise. The use of the pleonastic term "pairwise" helps to avoid confusion.
$endgroup$
– Umberto P.
42 mins ago
add a comment |
$begingroup$
More generally, sets are disjoint when their intersection is empty, but pairwise disjoint when any two of them are disjoint.
answered 55 mins ago
J.G.J.G.
29k22845
$endgroup$
add a comment |
$begingroup$
Consider the sets $A = {1,2}$, $B = {2,3}$, $C = {3, 1}$. Then $Acap Bcap C = varnothing$, but $A,B,C$ are not pairwise disjoint.
answered 55 mins ago
Kyle DuffyKyle Duffy
11
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Let $A={1,2}, B={2,3},C={3,4}$. Then the sets are disjoint because $Acap Bcap C=emptyset$, but not pairwise disjoint because you have pairs such as $A,B$ such that $Acap Bnot =emptyset$.
edited 54 mins ago
J.G.
29k22845
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
$endgroup$
$begingroup$
Rats! What is the notation for an empty set?
$endgroup$
– Oscar Lanzi
54 mins ago
$begingroup$
Thank you, @jg.
$endgroup$
– Oscar Lanzi
50 mins ago
add a comment |
Your Answer
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
${1,2},{2,3},{1,3}$ are disjoint but not pairwise disjoint.
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
$endgroup$
$begingroup$
Really? Who would call those disjoint sets?
$endgroup$
– John Lawrence Aspden
56 mins ago
1
$begingroup$
Everyone. Disjoint means their intersection is empty.
$endgroup$
– saulspatz
55 mins ago
$begingroup$
If that's true then I'll accept the answer (and thanks!). Can you cite or give a popular textbook that uses this definition?
$endgroup$
– John Lawrence Aspden
53 mins ago
$begingroup$
Sorry, I don't have any elementary textbooks any more.
$endgroup$
– saulspatz
50 mins ago
2
$begingroup$
If sets $A_1,A_2,dots,A_n$ are said to be disjoint then usually it is meant that the sets are pairwise disjoint. See Wolfram for instance. In the other case one says simply that the sets have an empty intersection.
$endgroup$
– drhab
32 mins ago
|
show 3 more comments
$begingroup$
${1,2},{2,3},{1,3}$ are disjoint but not pairwise disjoint.
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
$endgroup$
$begingroup$
Really? Who would call those disjoint sets?
$endgroup$
– John Lawrence Aspden
56 mins ago
1
$begingroup$
Everyone. Disjoint means their intersection is empty.
$endgroup$
– saulspatz
55 mins ago
$begingroup$
If that's true then I'll accept the answer (and thanks!). Can you cite or give a popular textbook that uses this definition?
$endgroup$
– John Lawrence Aspden
53 mins ago
$begingroup$
Sorry, I don't have any elementary textbooks any more.
$endgroup$
– saulspatz
50 mins ago
2
$begingroup$
If sets $A_1,A_2,dots,A_n$ are said to be disjoint then usually it is meant that the sets are pairwise disjoint. See Wolfram for instance. In the other case one says simply that the sets have an empty intersection.
$endgroup$
– drhab
32 mins ago
|
show 3 more comments
$begingroup$
${1,2},{2,3},{1,3}$ are disjoint but not pairwise disjoint.
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
$endgroup$
${1,2},{2,3},{1,3}$ are disjoint but not pairwise disjoint.
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
answered 57 mins ago
saulspatzsaulspatz
16.5k31332
16.5k31332
$begingroup$
Really? Who would call those disjoint sets?
$endgroup$
– John Lawrence Aspden
56 mins ago
1
$begingroup$
Everyone. Disjoint means their intersection is empty.
$endgroup$
– saulspatz
55 mins ago
$begingroup$
If that's true then I'll accept the answer (and thanks!). Can you cite or give a popular textbook that uses this definition?
$endgroup$
– John Lawrence Aspden
53 mins ago
$begingroup$
Sorry, I don't have any elementary textbooks any more.
$endgroup$
– saulspatz
50 mins ago
2
$begingroup$
If sets $A_1,A_2,dots,A_n$ are said to be disjoint then usually it is meant that the sets are pairwise disjoint. See Wolfram for instance. In the other case one says simply that the sets have an empty intersection.
$endgroup$
– drhab
32 mins ago
|
show 3 more comments
$begingroup$
Really? Who would call those disjoint sets?
$endgroup$
– John Lawrence Aspden
56 mins ago
1
$begingroup$
Everyone. Disjoint means their intersection is empty.
$endgroup$
– saulspatz
55 mins ago
$begingroup$
If that's true then I'll accept the answer (and thanks!). Can you cite or give a popular textbook that uses this definition?
$endgroup$
– John Lawrence Aspden
53 mins ago
$begingroup$
Sorry, I don't have any elementary textbooks any more.
$endgroup$
– saulspatz
50 mins ago
2
$begingroup$
If sets $A_1,A_2,dots,A_n$ are said to be disjoint then usually it is meant that the sets are pairwise disjoint. See Wolfram for instance. In the other case one says simply that the sets have an empty intersection.
$endgroup$
– drhab
32 mins ago
$begingroup$
Really? Who would call those disjoint sets?
$endgroup$
– John Lawrence Aspden
56 mins ago
$begingroup$
Really? Who would call those disjoint sets?
$endgroup$
– John Lawrence Aspden
56 mins ago
1
1
$begingroup$
Everyone. Disjoint means their intersection is empty.
$endgroup$
– saulspatz
55 mins ago
$begingroup$
Everyone. Disjoint means their intersection is empty.
$endgroup$
– saulspatz
55 mins ago
$begingroup$
If that's true then I'll accept the answer (and thanks!). Can you cite or give a popular textbook that uses this definition?
$endgroup$
– John Lawrence Aspden
53 mins ago
$begingroup$
If that's true then I'll accept the answer (and thanks!). Can you cite or give a popular textbook that uses this definition?
$endgroup$
– John Lawrence Aspden
53 mins ago
$begingroup$
Sorry, I don't have any elementary textbooks any more.
$endgroup$
– saulspatz
50 mins ago
$begingroup$
Sorry, I don't have any elementary textbooks any more.
$endgroup$
– saulspatz
50 mins ago
2
2
$begingroup$
If sets $A_1,A_2,dots,A_n$ are said to be disjoint then usually it is meant that the sets are pairwise disjoint. See Wolfram for instance. In the other case one says simply that the sets have an empty intersection.
$endgroup$
– drhab
32 mins ago
$begingroup$
If sets $A_1,A_2,dots,A_n$ are said to be disjoint then usually it is meant that the sets are pairwise disjoint. See Wolfram for instance. In the other case one says simply that the sets have an empty intersection.
$endgroup$
– drhab
32 mins ago
|
show 3 more comments
$begingroup$
In this context disjoint means $A cap B cap C = emptyset$.
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
$endgroup$
3
$begingroup$
Is that a standard meaning? I never saw the term formally defined that way in four years of undergrad math classes.
$endgroup$
– Connor Harris
56 mins ago
$begingroup$
me neither, but four people have answered the question this way in four minutes!
$endgroup$
– John Lawrence Aspden
51 mins ago
$begingroup$
If you define "disjoint" to mean "empty intersection" (which is the standard definition) then formally for a family of sets "disjoint" would mean the intersection of the entire family is empty unless stated otherwise. The use of the pleonastic term "pairwise" helps to avoid confusion.
$endgroup$
– Umberto P.
42 mins ago
add a comment |
$begingroup$
In this context disjoint means $A cap B cap C = emptyset$.
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
$endgroup$
3
$begingroup$
Is that a standard meaning? I never saw the term formally defined that way in four years of undergrad math classes.
$endgroup$
– Connor Harris
56 mins ago
$begingroup$
me neither, but four people have answered the question this way in four minutes!
$endgroup$
– John Lawrence Aspden
51 mins ago
$begingroup$
If you define "disjoint" to mean "empty intersection" (which is the standard definition) then formally for a family of sets "disjoint" would mean the intersection of the entire family is empty unless stated otherwise. The use of the pleonastic term "pairwise" helps to avoid confusion.
$endgroup$
– Umberto P.
42 mins ago
add a comment |
$begingroup$
In this context disjoint means $A cap B cap C = emptyset$.
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
$endgroup$
In this context disjoint means $A cap B cap C = emptyset$.
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
answered 57 mins ago
Umberto P.Umberto P.
39.7k13267
39.7k13267
3
$begingroup$
Is that a standard meaning? I never saw the term formally defined that way in four years of undergrad math classes.
$endgroup$
– Connor Harris
56 mins ago
$begingroup$
me neither, but four people have answered the question this way in four minutes!
$endgroup$
– John Lawrence Aspden
51 mins ago
$begingroup$
If you define "disjoint" to mean "empty intersection" (which is the standard definition) then formally for a family of sets "disjoint" would mean the intersection of the entire family is empty unless stated otherwise. The use of the pleonastic term "pairwise" helps to avoid confusion.
$endgroup$
– Umberto P.
42 mins ago
add a comment |
3
$begingroup$
Is that a standard meaning? I never saw the term formally defined that way in four years of undergrad math classes.
$endgroup$
– Connor Harris
56 mins ago
$begingroup$
me neither, but four people have answered the question this way in four minutes!
$endgroup$
– John Lawrence Aspden
51 mins ago
$begingroup$
If you define "disjoint" to mean "empty intersection" (which is the standard definition) then formally for a family of sets "disjoint" would mean the intersection of the entire family is empty unless stated otherwise. The use of the pleonastic term "pairwise" helps to avoid confusion.
$endgroup$
– Umberto P.
42 mins ago
3
3
$begingroup$
Is that a standard meaning? I never saw the term formally defined that way in four years of undergrad math classes.
$endgroup$
– Connor Harris
56 mins ago
$begingroup$
Is that a standard meaning? I never saw the term formally defined that way in four years of undergrad math classes.
$endgroup$
– Connor Harris
56 mins ago
$begingroup$
me neither, but four people have answered the question this way in four minutes!
$endgroup$
– John Lawrence Aspden
51 mins ago
$begingroup$
me neither, but four people have answered the question this way in four minutes!
$endgroup$
– John Lawrence Aspden
51 mins ago
$begingroup$
If you define "disjoint" to mean "empty intersection" (which is the standard definition) then formally for a family of sets "disjoint" would mean the intersection of the entire family is empty unless stated otherwise. The use of the pleonastic term "pairwise" helps to avoid confusion.
$endgroup$
– Umberto P.
42 mins ago
$begingroup$
If you define "disjoint" to mean "empty intersection" (which is the standard definition) then formally for a family of sets "disjoint" would mean the intersection of the entire family is empty unless stated otherwise. The use of the pleonastic term "pairwise" helps to avoid confusion.
$endgroup$
– Umberto P.
42 mins ago
add a comment |
$begingroup$
More generally, sets are disjoint when their intersection is empty, but pairwise disjoint when any two of them are disjoint.
answered 55 mins ago
J.G.J.G.
29k22845
$endgroup$
add a comment |
$begingroup$
More generally, sets are disjoint when their intersection is empty, but pairwise disjoint when any two of them are disjoint.
answered 55 mins ago
J.G.J.G.
29k22845
$endgroup$
add a comment |
$begingroup$
More generally, sets are disjoint when their intersection is empty, but pairwise disjoint when any two of them are disjoint.
answered 55 mins ago
J.G.J.G.
29k22845
$endgroup$
More generally, sets are disjoint when their intersection is empty, but pairwise disjoint when any two of them are disjoint.
answered 55 mins ago
J.G.J.G.
29k22845
answered 55 mins ago
J.G.J.G.
29k22845
answered 55 mins ago
J.G.J.G.
29k22845
answered 55 mins ago
J.G.J.G.
29k22845
29k22845
add a comment |
add a comment |
$begingroup$
Consider the sets $A = {1,2}$, $B = {2,3}$, $C = {3, 1}$. Then $Acap Bcap C = varnothing$, but $A,B,C$ are not pairwise disjoint.
answered 55 mins ago
Kyle DuffyKyle Duffy
11
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Consider the sets $A = {1,2}$, $B = {2,3}$, $C = {3, 1}$. Then $Acap Bcap C = varnothing$, but $A,B,C$ are not pairwise disjoint.
answered 55 mins ago
Kyle DuffyKyle Duffy
11
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
Consider the sets $A = {1,2}$, $B = {2,3}$, $C = {3, 1}$. Then $Acap Bcap C = varnothing$, but $A,B,C$ are not pairwise disjoint.
answered 55 mins ago
Kyle DuffyKyle Duffy
11
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
Consider the sets $A = {1,2}$, $B = {2,3}$, $C = {3, 1}$. Then $Acap Bcap C = varnothing$, but $A,B,C$ are not pairwise disjoint.
answered 55 mins ago
Kyle DuffyKyle Duffy
11
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 55 mins ago
Kyle DuffyKyle Duffy
11
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
answered 55 mins ago
Kyle DuffyKyle Duffy
11
answered 55 mins ago
Kyle DuffyKyle Duffy
11
11
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Kyle Duffy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
$begingroup$
Let $A={1,2}, B={2,3},C={3,4}$. Then the sets are disjoint because $Acap Bcap C=emptyset$, but not pairwise disjoint because you have pairs such as $A,B$ such that $Acap Bnot =emptyset$.
edited 54 mins ago
J.G.
29k22845
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
$endgroup$
$begingroup$
Rats! What is the notation for an empty set?
$endgroup$
– Oscar Lanzi
54 mins ago
$begingroup$
Thank you, @jg.
$endgroup$
– Oscar Lanzi
50 mins ago
add a comment |
$begingroup$
Let $A={1,2}, B={2,3},C={3,4}$. Then the sets are disjoint because $Acap Bcap C=emptyset$, but not pairwise disjoint because you have pairs such as $A,B$ such that $Acap Bnot =emptyset$.
edited 54 mins ago
J.G.
29k22845
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
$endgroup$
$begingroup$
Rats! What is the notation for an empty set?
$endgroup$
– Oscar Lanzi
54 mins ago
$begingroup$
Thank you, @jg.
$endgroup$
– Oscar Lanzi
50 mins ago
add a comment |
$begingroup$
Let $A={1,2}, B={2,3},C={3,4}$. Then the sets are disjoint because $Acap Bcap C=emptyset$, but not pairwise disjoint because you have pairs such as $A,B$ such that $Acap Bnot =emptyset$.
edited 54 mins ago
J.G.
29k22845
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
$endgroup$
Let $A={1,2}, B={2,3},C={3,4}$. Then the sets are disjoint because $Acap Bcap C=emptyset$, but not pairwise disjoint because you have pairs such as $A,B$ such that $Acap Bnot =emptyset$.
edited 54 mins ago
J.G.
29k22845
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
edited 54 mins ago
J.G.
29k22845
edited 54 mins ago
J.G.
29k22845
edited 54 mins ago
J.G.
29k22845
29k22845
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
answered 55 mins ago
Oscar LanziOscar Lanzi
13k12136
13k12136
$begingroup$
Rats! What is the notation for an empty set?
$endgroup$
– Oscar Lanzi
54 mins ago
$begingroup$
Thank you, @jg.
$endgroup$
– Oscar Lanzi
50 mins ago
add a comment |
$begingroup$
Rats! What is the notation for an empty set?
$endgroup$
– Oscar Lanzi
54 mins ago
$begingroup$
Thank you, @jg.
$endgroup$
– Oscar Lanzi
50 mins ago
$begingroup$
Rats! What is the notation for an empty set?
$endgroup$
– Oscar Lanzi
54 mins ago
$begingroup$
Rats! What is the notation for an empty set?
$endgroup$
– Oscar Lanzi
54 mins ago
$begingroup$
Thank you, @jg.
$endgroup$
– Oscar Lanzi
50 mins ago
$begingroup$
Thank you, @jg.
$endgroup$
– Oscar Lanzi
50 mins ago
add a comment |
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