Notation for subset of a set $A$ that is disjoint with every other set but $A$
Clash Royale CLAN TAG#URR8PPP
up vote
4
down vote
favorite
I have found myself making my own notation when considering several sets, say $A,B,C,D,E$, and wanting to denote for example the set $A^triangle = x: xin A land xnotin Y, Yneq X$. I know when considering just two sets like $A$ and $B$ I could write the somewhat more inconvenient $Asetminus B$, but I don't know how to conveniently write this when considering several sets. Is there a simple notation for $A^triangle$?
edit: Maybe I expressed myself unclearly. Although I appreciate the help, I don't consider $Asetminus(Bcup C cup D cup E)$ a simple way of writing $A^triangle$. It becomes very unpractical when having to write it over and over again, not to speak of when considering more sets than five. I'm looking for something similar to $A^triangle$, preferably just one small extra symbol denoting the set. Similar to how $barA$ can denote the complement of $A$. Again, sorry for not making this clear enough.
elementary-set-theory
asked Aug 26 at 20:36
DancingIceCream
385
add a comment |Â
up vote
4
down vote
favorite
I have found myself making my own notation when considering several sets, say $A,B,C,D,E$, and wanting to denote for example the set $A^triangle = x: xin A land xnotin Y, Yneq X$. I know when considering just two sets like $A$ and $B$ I could write the somewhat more inconvenient $Asetminus B$, but I don't know how to conveniently write this when considering several sets. Is there a simple notation for $A^triangle$?
edit: Maybe I expressed myself unclearly. Although I appreciate the help, I don't consider $Asetminus(Bcup C cup D cup E)$ a simple way of writing $A^triangle$. It becomes very unpractical when having to write it over and over again, not to speak of when considering more sets than five. I'm looking for something similar to $A^triangle$, preferably just one small extra symbol denoting the set. Similar to how $barA$ can denote the complement of $A$. Again, sorry for not making this clear enough.
elementary-set-theory
asked Aug 26 at 20:36
DancingIceCream
385
3
Would $A setminus (B cup C cup D cup E)$ work?
– Evan Chen
Aug 26 at 21:31
@EvanChen I'm afraid I expressed myself a bit unlcearly, I don't find that notation simple or convenient. I thought it was clear by me bringing up the example $Asetminus B$. But thank you for your help.
– DancingIceCream
Aug 27 at 5:17
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I have found myself making my own notation when considering several sets, say $A,B,C,D,E$, and wanting to denote for example the set $A^triangle = x: xin A land xnotin Y, Yneq X$. I know when considering just two sets like $A$ and $B$ I could write the somewhat more inconvenient $Asetminus B$, but I don't know how to conveniently write this when considering several sets. Is there a simple notation for $A^triangle$?
edit: Maybe I expressed myself unclearly. Although I appreciate the help, I don't consider $Asetminus(Bcup C cup D cup E)$ a simple way of writing $A^triangle$. It becomes very unpractical when having to write it over and over again, not to speak of when considering more sets than five. I'm looking for something similar to $A^triangle$, preferably just one small extra symbol denoting the set. Similar to how $barA$ can denote the complement of $A$. Again, sorry for not making this clear enough.
elementary-set-theory
asked Aug 26 at 20:36
DancingIceCream
385
I have found myself making my own notation when considering several sets, say $A,B,C,D,E$, and wanting to denote for example the set $A^triangle = x: xin A land xnotin Y, Yneq X$. I know when considering just two sets like $A$ and $B$ I could write the somewhat more inconvenient $Asetminus B$, but I don't know how to conveniently write this when considering several sets. Is there a simple notation for $A^triangle$?
edit: Maybe I expressed myself unclearly. Although I appreciate the help, I don't consider $Asetminus(Bcup C cup D cup E)$ a simple way of writing $A^triangle$. It becomes very unpractical when having to write it over and over again, not to speak of when considering more sets than five. I'm looking for something similar to $A^triangle$, preferably just one small extra symbol denoting the set. Similar to how $barA$ can denote the complement of $A$. Again, sorry for not making this clear enough.
elementary-set-theory
asked Aug 26 at 20:36
DancingIceCream
385
edited Aug 27 at 5:53
asked Aug 26 at 20:36
DancingIceCream
385
asked Aug 26 at 20:36
DancingIceCream
385
asked Aug 26 at 20:36
DancingIceCream
385
385
3
Would $A setminus (B cup C cup D cup E)$ work?
– Evan Chen
Aug 26 at 21:31
@EvanChen I'm afraid I expressed myself a bit unlcearly, I don't find that notation simple or convenient. I thought it was clear by me bringing up the example $Asetminus B$. But thank you for your help.
– DancingIceCream
Aug 27 at 5:17
add a comment |Â
3
Would $A setminus (B cup C cup D cup E)$ work?
– Evan Chen
Aug 26 at 21:31
@EvanChen I'm afraid I expressed myself a bit unlcearly, I don't find that notation simple or convenient. I thought it was clear by me bringing up the example $Asetminus B$. But thank you for your help.
– DancingIceCream
Aug 27 at 5:17
3
3
Would $A setminus (B cup C cup D cup E)$ work?
– Evan Chen
Aug 26 at 21:31
Would $A setminus (B cup C cup D cup E)$ work?
– Evan Chen
Aug 26 at 21:31
@EvanChen I'm afraid I expressed myself a bit unlcearly, I don't find that notation simple or convenient. I thought it was clear by me bringing up the example $Asetminus B$. But thank you for your help.
– DancingIceCream
Aug 27 at 5:17
@EvanChen I'm afraid I expressed myself a bit unlcearly, I don't find that notation simple or convenient. I thought it was clear by me bringing up the example $Asetminus B$. But thank you for your help.
– DancingIceCream
Aug 27 at 5:17
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
0
down vote
You can find out that $x in A^△Leftrightarrow (x in A ∧x notin B ∧x notin C∧x notin D∧x notin E)$.
Hence, by de Morgan law, $$x in A^△Leftrightarrow (x in A ∧neg (x in B vee x in Cvee x in Dvee x in E))$$$$Leftrightarrow (x in A ∧neg (x in Bcup Ccup D cup E ))$$$$Leftrightarrow (x in Asetminus( Bcup Ccup D cup E ))$$
Therefore, $Asetminus( Bcup Ccup D cup E )$ is what you are looking for.
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
add a comment |Â
up vote
0
down vote
We have already standard notations of difference and union that cover your wanted new notation.
If your set is only non-disjoint with $A$ and disjoint with some family of sets $F$ we could do some shortcuts and write (if B is designation of that set) that as $(B cap A neq emptyset) land (B cap (bigcup F) = emptyset) $
We could write that also as $displaystyle bigcap_B (A,F)=( supset emptyset , = emptyset) $
answered Aug 27 at 5:47
Right
1,046213
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
You can find out that $x in A^△Leftrightarrow (x in A ∧x notin B ∧x notin C∧x notin D∧x notin E)$.
Hence, by de Morgan law, $$x in A^△Leftrightarrow (x in A ∧neg (x in B vee x in Cvee x in Dvee x in E))$$$$Leftrightarrow (x in A ∧neg (x in Bcup Ccup D cup E ))$$$$Leftrightarrow (x in Asetminus( Bcup Ccup D cup E ))$$
Therefore, $Asetminus( Bcup Ccup D cup E )$ is what you are looking for.
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
add a comment |Â
up vote
0
down vote
You can find out that $x in A^△Leftrightarrow (x in A ∧x notin B ∧x notin C∧x notin D∧x notin E)$.
Hence, by de Morgan law, $$x in A^△Leftrightarrow (x in A ∧neg (x in B vee x in Cvee x in Dvee x in E))$$$$Leftrightarrow (x in A ∧neg (x in Bcup Ccup D cup E ))$$$$Leftrightarrow (x in Asetminus( Bcup Ccup D cup E ))$$
Therefore, $Asetminus( Bcup Ccup D cup E )$ is what you are looking for.
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
add a comment |Â
up vote
0
down vote
up vote
0
down vote
You can find out that $x in A^△Leftrightarrow (x in A ∧x notin B ∧x notin C∧x notin D∧x notin E)$.
Hence, by de Morgan law, $$x in A^△Leftrightarrow (x in A ∧neg (x in B vee x in Cvee x in Dvee x in E))$$$$Leftrightarrow (x in A ∧neg (x in Bcup Ccup D cup E ))$$$$Leftrightarrow (x in Asetminus( Bcup Ccup D cup E ))$$
Therefore, $Asetminus( Bcup Ccup D cup E )$ is what you are looking for.
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
You can find out that $x in A^△Leftrightarrow (x in A ∧x notin B ∧x notin C∧x notin D∧x notin E)$.
Hence, by de Morgan law, $$x in A^△Leftrightarrow (x in A ∧neg (x in B vee x in Cvee x in Dvee x in E))$$$$Leftrightarrow (x in A ∧neg (x in Bcup Ccup D cup E ))$$$$Leftrightarrow (x in Asetminus( Bcup Ccup D cup E ))$$
Therefore, $Asetminus( Bcup Ccup D cup E )$ is what you are looking for.
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
answered Aug 27 at 0:13
Steve Cheng é„Â宗弘
236
236
add a comment |Â
add a comment |Â
up vote
0
down vote
We have already standard notations of difference and union that cover your wanted new notation.
If your set is only non-disjoint with $A$ and disjoint with some family of sets $F$ we could do some shortcuts and write (if B is designation of that set) that as $(B cap A neq emptyset) land (B cap (bigcup F) = emptyset) $
We could write that also as $displaystyle bigcap_B (A,F)=( supset emptyset , = emptyset) $
answered Aug 27 at 5:47
Right
1,046213
add a comment |Â
up vote
0
down vote
We have already standard notations of difference and union that cover your wanted new notation.
If your set is only non-disjoint with $A$ and disjoint with some family of sets $F$ we could do some shortcuts and write (if B is designation of that set) that as $(B cap A neq emptyset) land (B cap (bigcup F) = emptyset) $
We could write that also as $displaystyle bigcap_B (A,F)=( supset emptyset , = emptyset) $
answered Aug 27 at 5:47
Right
1,046213
add a comment |Â
up vote
0
down vote
up vote
0
down vote
We have already standard notations of difference and union that cover your wanted new notation.
If your set is only non-disjoint with $A$ and disjoint with some family of sets $F$ we could do some shortcuts and write (if B is designation of that set) that as $(B cap A neq emptyset) land (B cap (bigcup F) = emptyset) $
We could write that also as $displaystyle bigcap_B (A,F)=( supset emptyset , = emptyset) $
answered Aug 27 at 5:47
Right
1,046213
We have already standard notations of difference and union that cover your wanted new notation.
If your set is only non-disjoint with $A$ and disjoint with some family of sets $F$ we could do some shortcuts and write (if B is designation of that set) that as $(B cap A neq emptyset) land (B cap (bigcup F) = emptyset) $
We could write that also as $displaystyle bigcap_B (A,F)=( supset emptyset , = emptyset) $
answered Aug 27 at 5:47
Right
1,046213
answered Aug 27 at 5:47
Right
1,046213
answered Aug 27 at 5:47
Right
1,046213
answered Aug 27 at 5:47
Right
1,046213
1,046213
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%2f2895491%2fnotation-for-subset-of-a-set-a-that-is-disjoint-with-every-other-set-but-a%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
3
Would $A setminus (B cup C cup D cup E)$ work?
– Evan Chen
Aug 26 at 21:31
@EvanChen I'm afraid I expressed myself a bit unlcearly, I don't find that notation simple or convenient. I thought it was clear by me bringing up the example $Asetminus B$. But thank you for your help.
– DancingIceCream
Aug 27 at 5:17