Reflexive property of relations? [duplicate]
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This question already has an answer here:
Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
3 answers
If a relation is both symmetric and anti-symmetric, isn't it always reflexive? I am supposed to consider a set, S, where some Relation A is both symmetric and anti-symmetric. I thought that anti-symmetric means that if (x, y) is in the relation, then (y, x) cannot be in it unless x = y. So, wouldn't it hold that it is reflexive?
relations
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
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marked as duplicate by Brian Borchers, Misha Lavrov, Chinnapparaj R, Lord Shark the Unknown, Leucippus Dec 1 '18 at 7:14
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
3 answers
If a relation is both symmetric and anti-symmetric, isn't it always reflexive? I am supposed to consider a set, S, where some Relation A is both symmetric and anti-symmetric. I thought that anti-symmetric means that if (x, y) is in the relation, then (y, x) cannot be in it unless x = y. So, wouldn't it hold that it is reflexive?
relations
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
$endgroup$
marked as duplicate by Brian Borchers, Misha Lavrov, Chinnapparaj R, Lord Shark the Unknown, Leucippus Dec 1 '18 at 7:14
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
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let $A={1,2}$ and the relation be ${(1,1)}$. this is both symmetric and anti-symmetric but not reflexive.
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– Anurag A
Dec 1 '18 at 1:28
1
$begingroup$
An empty relation is symmetric and anti-symmetric. Any sub-relation of the equality relation is both symmetric and anti-symmetric, and these are the only such relations.
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– Thomas Andrews
Dec 1 '18 at 1:35
add a comment |
$begingroup$
This question already has an answer here:
Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
3 answers
If a relation is both symmetric and anti-symmetric, isn't it always reflexive? I am supposed to consider a set, S, where some Relation A is both symmetric and anti-symmetric. I thought that anti-symmetric means that if (x, y) is in the relation, then (y, x) cannot be in it unless x = y. So, wouldn't it hold that it is reflexive?
relations
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
$endgroup$
This question already has an answer here:
Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
3 answers
If a relation is both symmetric and anti-symmetric, isn't it always reflexive? I am supposed to consider a set, S, where some Relation A is both symmetric and anti-symmetric. I thought that anti-symmetric means that if (x, y) is in the relation, then (y, x) cannot be in it unless x = y. So, wouldn't it hold that it is reflexive?
This question already has an answer here:
Whats the difference between Antisymmetric and reflexive? (Set Theory/Discrete math)
3 answers
relations
relations
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
asked Dec 1 '18 at 1:25
Justin DeeJustin Dee
615
615
marked as duplicate by Brian Borchers, Misha Lavrov, Chinnapparaj R, Lord Shark the Unknown, Leucippus Dec 1 '18 at 7:14
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Brian Borchers, Misha Lavrov, Chinnapparaj R, Lord Shark the Unknown, Leucippus Dec 1 '18 at 7:14
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
1
$begingroup$
let $A={1,2}$ and the relation be ${(1,1)}$. this is both symmetric and anti-symmetric but not reflexive.
$endgroup$
– Anurag A
Dec 1 '18 at 1:28
1
$begingroup$
An empty relation is symmetric and anti-symmetric. Any sub-relation of the equality relation is both symmetric and anti-symmetric, and these are the only such relations.
$endgroup$
– Thomas Andrews
Dec 1 '18 at 1:35
add a comment |
1
$begingroup$
let $A={1,2}$ and the relation be ${(1,1)}$. this is both symmetric and anti-symmetric but not reflexive.
$endgroup$
– Anurag A
Dec 1 '18 at 1:28
1
$begingroup$
An empty relation is symmetric and anti-symmetric. Any sub-relation of the equality relation is both symmetric and anti-symmetric, and these are the only such relations.
$endgroup$
– Thomas Andrews
Dec 1 '18 at 1:35
1
1
$begingroup$
let $A={1,2}$ and the relation be ${(1,1)}$. this is both symmetric and anti-symmetric but not reflexive.
$endgroup$
– Anurag A
Dec 1 '18 at 1:28
$begingroup$
let $A={1,2}$ and the relation be ${(1,1)}$. this is both symmetric and anti-symmetric but not reflexive.
$endgroup$
– Anurag A
Dec 1 '18 at 1:28
1
1
$begingroup$
An empty relation is symmetric and anti-symmetric. Any sub-relation of the equality relation is both symmetric and anti-symmetric, and these are the only such relations.
$endgroup$
– Thomas Andrews
Dec 1 '18 at 1:35
$begingroup$
An empty relation is symmetric and anti-symmetric. Any sub-relation of the equality relation is both symmetric and anti-symmetric, and these are the only such relations.
$endgroup$
– Thomas Andrews
Dec 1 '18 at 1:35
add a comment |
1 Answer
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Reflexive means for all elements must be related to themselves but that other elements may (or may not) be related to each other.
Both Symmetric and anti-symmetric means elements can only be related to the themselves (thought they don't need to be) and cant be related to any other elements[1].
So a relation that is symmetric and anti-symmetric need not be reflexive if not all elements are related to themselves. (This would mean those elements are not related to anything.)
Example: If you have a set $S={1,2}$ and $1 sim 1$ but $2$ is not related to anything that is symmetric. (As every case $a sim b$ then $a =b = 1$ and $b sim 1$.) And antismmetric. (As $asim b$ and $bsim a$ means $a = b =1$). But it is not reflexive as $2not sim 2$.
[1]. Symmetric means if $a sim b$ then $b sim a$. Anti-symmetry means if $a sim b$ and $bsim a$ then $a =b$. If a relation is both then $a sim b implies bsim a implies a = b$.
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Reflexive means for all elements must be related to themselves but that other elements may (or may not) be related to each other.
Both Symmetric and anti-symmetric means elements can only be related to the themselves (thought they don't need to be) and cant be related to any other elements[1].
So a relation that is symmetric and anti-symmetric need not be reflexive if not all elements are related to themselves. (This would mean those elements are not related to anything.)
Example: If you have a set $S={1,2}$ and $1 sim 1$ but $2$ is not related to anything that is symmetric. (As every case $a sim b$ then $a =b = 1$ and $b sim 1$.) And antismmetric. (As $asim b$ and $bsim a$ means $a = b =1$). But it is not reflexive as $2not sim 2$.
[1]. Symmetric means if $a sim b$ then $b sim a$. Anti-symmetry means if $a sim b$ and $bsim a$ then $a =b$. If a relation is both then $a sim b implies bsim a implies a = b$.
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
$endgroup$
add a comment |
$begingroup$
Reflexive means for all elements must be related to themselves but that other elements may (or may not) be related to each other.
Both Symmetric and anti-symmetric means elements can only be related to the themselves (thought they don't need to be) and cant be related to any other elements[1].
So a relation that is symmetric and anti-symmetric need not be reflexive if not all elements are related to themselves. (This would mean those elements are not related to anything.)
Example: If you have a set $S={1,2}$ and $1 sim 1$ but $2$ is not related to anything that is symmetric. (As every case $a sim b$ then $a =b = 1$ and $b sim 1$.) And antismmetric. (As $asim b$ and $bsim a$ means $a = b =1$). But it is not reflexive as $2not sim 2$.
[1]. Symmetric means if $a sim b$ then $b sim a$. Anti-symmetry means if $a sim b$ and $bsim a$ then $a =b$. If a relation is both then $a sim b implies bsim a implies a = b$.
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
$endgroup$
add a comment |
$begingroup$
Reflexive means for all elements must be related to themselves but that other elements may (or may not) be related to each other.
Both Symmetric and anti-symmetric means elements can only be related to the themselves (thought they don't need to be) and cant be related to any other elements[1].
So a relation that is symmetric and anti-symmetric need not be reflexive if not all elements are related to themselves. (This would mean those elements are not related to anything.)
Example: If you have a set $S={1,2}$ and $1 sim 1$ but $2$ is not related to anything that is symmetric. (As every case $a sim b$ then $a =b = 1$ and $b sim 1$.) And antismmetric. (As $asim b$ and $bsim a$ means $a = b =1$). But it is not reflexive as $2not sim 2$.
[1]. Symmetric means if $a sim b$ then $b sim a$. Anti-symmetry means if $a sim b$ and $bsim a$ then $a =b$. If a relation is both then $a sim b implies bsim a implies a = b$.
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
$endgroup$
Reflexive means for all elements must be related to themselves but that other elements may (or may not) be related to each other.
Both Symmetric and anti-symmetric means elements can only be related to the themselves (thought they don't need to be) and cant be related to any other elements[1].
So a relation that is symmetric and anti-symmetric need not be reflexive if not all elements are related to themselves. (This would mean those elements are not related to anything.)
Example: If you have a set $S={1,2}$ and $1 sim 1$ but $2$ is not related to anything that is symmetric. (As every case $a sim b$ then $a =b = 1$ and $b sim 1$.) And antismmetric. (As $asim b$ and $bsim a$ means $a = b =1$). But it is not reflexive as $2not sim 2$.
[1]. Symmetric means if $a sim b$ then $b sim a$. Anti-symmetry means if $a sim b$ and $bsim a$ then $a =b$. If a relation is both then $a sim b implies bsim a implies a = b$.
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
answered Dec 1 '18 at 5:14
fleabloodfleablood
69.2k22685
69.2k22685
add a comment |
add a comment |
1
$begingroup$
let $A={1,2}$ and the relation be ${(1,1)}$. this is both symmetric and anti-symmetric but not reflexive.
$endgroup$
– Anurag A
Dec 1 '18 at 1:28
1
$begingroup$
An empty relation is symmetric and anti-symmetric. Any sub-relation of the equality relation is both symmetric and anti-symmetric, and these are the only such relations.
$endgroup$
– Thomas Andrews
Dec 1 '18 at 1:35