Not intersection of two sets

Question

I am trying to figure out the word/operation to what is not in the intersection of two sets.

How I am going about this now is $(\{A\} \cup \{B\}) - (\{A\} \cap \{B\})$

Is there a better way to go about this?

B is the intersection? You mean &? Do you mean A,B as sets or as singletons?
This is called the symmetric difference, BTW. But that's just a name. It's not clear from your question what "better" means, what kinds of sets you're working with, what kinds of operations you can do efficiently, etc.
Are you sure you mean $\{A\}\cup\{B\}$ and $\{A\}\cap\{B\}$, and not $A\cup B$ and $A\cap B$? If you mean what you wrote, then you get either $\emptyset$ if $A=B$, and $\{A,B\}$ if $A\neq B$. If you did not mean to have the curly brackets, you want the symmetric difference, which is often denoted by $A\Delta B$.
There are many alternative ways of writing the symmetric difference (when you don't have those curly brackets). E.g., for any two sets $A$ and $B$, $$(A\cup B)-(A\cap B) = (A-B)\cup(B-A) = (A\cap B^c)\cup (B\cap A^c).$$

trutheality · Accepted Answer · 2011-06-27 20:29:53Z

up vote 5 down vote accepted

$$(A \cup B) - (A \cap B)$$ Is called the symmetric difference and is often denoted by $$ A \Delta B $$

It is analogous to the logical "exclusive or" operation (as union is analogous to "or" and intersection is analogous to "and").

answered Jun 27 '11 at 20:29

community wiki

It's frequently shortened to xor (or XOR). – Josh Jun 28 '11 at 1:00

user9352 · Answer 2 · 2011-06-27 20:28:28Z

up vote 1 down vote

$(A \cap B^c)\cup (B\cap A^c)$

answered Jun 27 '11 at 20:28

user9352
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2 Answers