What is symmetric about set symmetric difference?
Clash Royale CLAN TAG#URR8PPP
up vote
9
down vote
favorite
Using $Delta$ for set symmetric difference,
$A Delta B$ is all the elements in exactly one of the sets but not all of them.
$A Delta B Delta C $ is all the elements in exactly one of the sets or all of them.
I appreciate there is an even number of sets in the first example and an odd number in the second (and associativity implies no order ambiguity), but what is symmetric about set symmetric difference?
elementary-set-theory
asked 13 hours ago
PM.
3,2432823
add a comment |Â
up vote
9
down vote
favorite
Using $Delta$ for set symmetric difference,
$A Delta B$ is all the elements in exactly one of the sets but not all of them.
$A Delta B Delta C $ is all the elements in exactly one of the sets or all of them.
I appreciate there is an even number of sets in the first example and an odd number in the second (and associativity implies no order ambiguity), but what is symmetric about set symmetric difference?
elementary-set-theory
asked 13 hours ago
PM.
3,2432823
add a comment |Â
up vote
9
down vote
favorite
up vote
9
down vote
favorite
Using $Delta$ for set symmetric difference,
$A Delta B$ is all the elements in exactly one of the sets but not all of them.
$A Delta B Delta C $ is all the elements in exactly one of the sets or all of them.
I appreciate there is an even number of sets in the first example and an odd number in the second (and associativity implies no order ambiguity), but what is symmetric about set symmetric difference?
elementary-set-theory
asked 13 hours ago
PM.
3,2432823
Using $Delta$ for set symmetric difference,
$A Delta B$ is all the elements in exactly one of the sets but not all of them.
$A Delta B Delta C $ is all the elements in exactly one of the sets or all of them.
I appreciate there is an even number of sets in the first example and an odd number in the second (and associativity implies no order ambiguity), but what is symmetric about set symmetric difference?
elementary-set-theory
elementary-set-theory
asked 13 hours ago
PM.
3,2432823
asked 13 hours ago
PM.
3,2432823
asked 13 hours ago
PM.
3,2432823
asked 13 hours ago
PM.
3,2432823
asked 13 hours ago
PM.
3,2432823
3,2432823
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
32
down vote
accepted
A function in two variables $f(x,y)$ is called symmetric if $f(x,y)=f(y,x)$.
It is easy to see that $AmathbintriangleB=BmathbintriangleA$, exactly because being in exactly in one of $A$ or $B$ is the same as being exactly in one of $B$ and $A$.
This is in contrast to set difference, where $Asetminus B$ is generally not the same as $Bsetminus A$.
answered 13 hours ago
Asaf Karagila♦
296k32414740
3
I'm curious - is there a specific reason why functional parlance is used (symmetric), rather than operator parlance (commutative)?
– Dancrumb
8 hours ago
6
An operator would be from $X^2to X$, whereas a function can be from $X^2to Y$ for some other $Y$.
– Asaf Karagila♦
8 hours ago
2
@AsafKaragila That does not really support the function instead of operator view for the nomenclature, as we can best view this as $Deltacolonmathcal P(U)^2to mathcal P(U)$ for some universal $U$. I understand $Delta$ as a symmetrized difference though, as it is the union over all differences of permutations of the arguments, $Asetminus Bcup Bsetminus A$
– Hagen von Eitzen
3 hours ago
@Hagen: Yes, the symmetric difference is a symmetric operator on sets. But the general definition is a "symmetric function".
– Asaf Karagila♦
2 hours ago
@Dancrumb note that operators are also sometimes called "symmetric" (or self-adjoint).
– user7530
2 hours ago
add a comment |Â
up vote
9
down vote
Here is a Venn diagram for normal set difference:
Here is a Venn diagram for symmetric set difference:
Note the symmetry of the latter, and asymmetry of the former.
answered 8 hours ago
Daniel R. Collins
5,6431533
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
32
down vote
accepted
A function in two variables $f(x,y)$ is called symmetric if $f(x,y)=f(y,x)$.
It is easy to see that $AmathbintriangleB=BmathbintriangleA$, exactly because being in exactly in one of $A$ or $B$ is the same as being exactly in one of $B$ and $A$.
This is in contrast to set difference, where $Asetminus B$ is generally not the same as $Bsetminus A$.
answered 13 hours ago
Asaf Karagila♦
296k32414740
3
I'm curious - is there a specific reason why functional parlance is used (symmetric), rather than operator parlance (commutative)?
– Dancrumb
8 hours ago
6
An operator would be from $X^2to X$, whereas a function can be from $X^2to Y$ for some other $Y$.
– Asaf Karagila♦
8 hours ago
2
@AsafKaragila That does not really support the function instead of operator view for the nomenclature, as we can best view this as $Deltacolonmathcal P(U)^2to mathcal P(U)$ for some universal $U$. I understand $Delta$ as a symmetrized difference though, as it is the union over all differences of permutations of the arguments, $Asetminus Bcup Bsetminus A$
– Hagen von Eitzen
3 hours ago
@Hagen: Yes, the symmetric difference is a symmetric operator on sets. But the general definition is a "symmetric function".
– Asaf Karagila♦
2 hours ago
@Dancrumb note that operators are also sometimes called "symmetric" (or self-adjoint).
– user7530
2 hours ago
add a comment |Â
up vote
32
down vote
accepted
A function in two variables $f(x,y)$ is called symmetric if $f(x,y)=f(y,x)$.
It is easy to see that $AmathbintriangleB=BmathbintriangleA$, exactly because being in exactly in one of $A$ or $B$ is the same as being exactly in one of $B$ and $A$.
This is in contrast to set difference, where $Asetminus B$ is generally not the same as $Bsetminus A$.
answered 13 hours ago
Asaf Karagila♦
296k32414740
3
I'm curious - is there a specific reason why functional parlance is used (symmetric), rather than operator parlance (commutative)?
– Dancrumb
8 hours ago
6
An operator would be from $X^2to X$, whereas a function can be from $X^2to Y$ for some other $Y$.
– Asaf Karagila♦
8 hours ago
2
@AsafKaragila That does not really support the function instead of operator view for the nomenclature, as we can best view this as $Deltacolonmathcal P(U)^2to mathcal P(U)$ for some universal $U$. I understand $Delta$ as a symmetrized difference though, as it is the union over all differences of permutations of the arguments, $Asetminus Bcup Bsetminus A$
– Hagen von Eitzen
3 hours ago
@Hagen: Yes, the symmetric difference is a symmetric operator on sets. But the general definition is a "symmetric function".
– Asaf Karagila♦
2 hours ago
@Dancrumb note that operators are also sometimes called "symmetric" (or self-adjoint).
– user7530
2 hours ago
add a comment |Â
up vote
32
down vote
accepted
up vote
32
down vote
accepted
A function in two variables $f(x,y)$ is called symmetric if $f(x,y)=f(y,x)$.
It is easy to see that $AmathbintriangleB=BmathbintriangleA$, exactly because being in exactly in one of $A$ or $B$ is the same as being exactly in one of $B$ and $A$.
This is in contrast to set difference, where $Asetminus B$ is generally not the same as $Bsetminus A$.
answered 13 hours ago
Asaf Karagila♦
296k32414740
A function in two variables $f(x,y)$ is called symmetric if $f(x,y)=f(y,x)$.
It is easy to see that $AmathbintriangleB=BmathbintriangleA$, exactly because being in exactly in one of $A$ or $B$ is the same as being exactly in one of $B$ and $A$.
This is in contrast to set difference, where $Asetminus B$ is generally not the same as $Bsetminus A$.
answered 13 hours ago
Asaf Karagila♦
296k32414740
answered 13 hours ago
Asaf Karagila♦
296k32414740
answered 13 hours ago
Asaf Karagila♦
296k32414740
answered 13 hours ago
Asaf Karagila♦
296k32414740
296k32414740
3
I'm curious - is there a specific reason why functional parlance is used (symmetric), rather than operator parlance (commutative)?
– Dancrumb
8 hours ago
6
An operator would be from $X^2to X$, whereas a function can be from $X^2to Y$ for some other $Y$.
– Asaf Karagila♦
8 hours ago
2
@AsafKaragila That does not really support the function instead of operator view for the nomenclature, as we can best view this as $Deltacolonmathcal P(U)^2to mathcal P(U)$ for some universal $U$. I understand $Delta$ as a symmetrized difference though, as it is the union over all differences of permutations of the arguments, $Asetminus Bcup Bsetminus A$
– Hagen von Eitzen
3 hours ago
@Hagen: Yes, the symmetric difference is a symmetric operator on sets. But the general definition is a "symmetric function".
– Asaf Karagila♦
2 hours ago
@Dancrumb note that operators are also sometimes called "symmetric" (or self-adjoint).
– user7530
2 hours ago
add a comment |Â
3
I'm curious - is there a specific reason why functional parlance is used (symmetric), rather than operator parlance (commutative)?
– Dancrumb
8 hours ago
6
An operator would be from $X^2to X$, whereas a function can be from $X^2to Y$ for some other $Y$.
– Asaf Karagila♦
8 hours ago
2
@AsafKaragila That does not really support the function instead of operator view for the nomenclature, as we can best view this as $Deltacolonmathcal P(U)^2to mathcal P(U)$ for some universal $U$. I understand $Delta$ as a symmetrized difference though, as it is the union over all differences of permutations of the arguments, $Asetminus Bcup Bsetminus A$
– Hagen von Eitzen
3 hours ago
@Hagen: Yes, the symmetric difference is a symmetric operator on sets. But the general definition is a "symmetric function".
– Asaf Karagila♦
2 hours ago
@Dancrumb note that operators are also sometimes called "symmetric" (or self-adjoint).
– user7530
2 hours ago
3
3
I'm curious - is there a specific reason why functional parlance is used (symmetric), rather than operator parlance (commutative)?
– Dancrumb
8 hours ago
I'm curious - is there a specific reason why functional parlance is used (symmetric), rather than operator parlance (commutative)?
– Dancrumb
8 hours ago
6
6
An operator would be from $X^2to X$, whereas a function can be from $X^2to Y$ for some other $Y$.
– Asaf Karagila♦
8 hours ago
An operator would be from $X^2to X$, whereas a function can be from $X^2to Y$ for some other $Y$.
– Asaf Karagila♦
8 hours ago
2
2
@AsafKaragila That does not really support the function instead of operator view for the nomenclature, as we can best view this as $Deltacolonmathcal P(U)^2to mathcal P(U)$ for some universal $U$. I understand $Delta$ as a symmetrized difference though, as it is the union over all differences of permutations of the arguments, $Asetminus Bcup Bsetminus A$
– Hagen von Eitzen
3 hours ago
@AsafKaragila That does not really support the function instead of operator view for the nomenclature, as we can best view this as $Deltacolonmathcal P(U)^2to mathcal P(U)$ for some universal $U$. I understand $Delta$ as a symmetrized difference though, as it is the union over all differences of permutations of the arguments, $Asetminus Bcup Bsetminus A$
– Hagen von Eitzen
3 hours ago
@Hagen: Yes, the symmetric difference is a symmetric operator on sets. But the general definition is a "symmetric function".
– Asaf Karagila♦
2 hours ago
@Hagen: Yes, the symmetric difference is a symmetric operator on sets. But the general definition is a "symmetric function".
– Asaf Karagila♦
2 hours ago
@Dancrumb note that operators are also sometimes called "symmetric" (or self-adjoint).
– user7530
2 hours ago
@Dancrumb note that operators are also sometimes called "symmetric" (or self-adjoint).
– user7530
2 hours ago
add a comment |Â
up vote
9
down vote
Here is a Venn diagram for normal set difference:
Here is a Venn diagram for symmetric set difference:
Note the symmetry of the latter, and asymmetry of the former.
answered 8 hours ago
Daniel R. Collins
5,6431533
add a comment |Â
up vote
9
down vote
Here is a Venn diagram for normal set difference:
Here is a Venn diagram for symmetric set difference:
Note the symmetry of the latter, and asymmetry of the former.
answered 8 hours ago
Daniel R. Collins
5,6431533
add a comment |Â
up vote
9
down vote
up vote
9
down vote
Here is a Venn diagram for normal set difference:
Here is a Venn diagram for symmetric set difference:
Note the symmetry of the latter, and asymmetry of the former.
answered 8 hours ago
Daniel R. Collins
5,6431533
Here is a Venn diagram for normal set difference:
Here is a Venn diagram for symmetric set difference:
Note the symmetry of the latter, and asymmetry of the former.
answered 8 hours ago
Daniel R. Collins
5,6431533
answered 8 hours ago
Daniel R. Collins
5,6431533
answered 8 hours ago
Daniel R. Collins
5,6431533
answered 8 hours ago
Daniel R. Collins
5,6431533
5,6431533
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2948351%2fwhat-is-symmetric-about-set-symmetric-difference%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
