# Relation between XOR and Symmetric difference

I noticed that XOR and symmetric difference use the same symbol, $\oplus$.

They also seem to have a similar structure:

XOR: $(\neg P\wedge Q)\vee(P\wedge \neg Q)$

Symmetric Difference: $(A\cap B^C)\cup(B\cap A^C)$

Is there a relation between them?

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Rewrite your definition of XOR to be $(P\wedge ¬Q)\vee(Q\wedge ¬P)$. Now do you see the relationship? – Chris Taylor Nov 21 '11 at 10:31
It is the same thing. Symmetric difference is XOR for sets, XOR is symmetric difference for truth values. Thinking of both as Boolean algebra operations then they are indeed the same. – Asaf Karagila Nov 21 '11 at 10:40

Yes, there is. Let $A\;\triangle\; B$ denote the symmetric difference of the sets $A$ and $B$. Given an object $x$, $$x\in A\;\triangle\; B\iff (x\in A)\text{ XOR }(x\in B).$$ In general, one has a correspondence between statements in set theory and statements in logic, e.g. $$x\in A\cup B\iff (x\in A)\text{ OR }(x\in B)$$ $$x\in A\cap B\iff (x\in A)\text{ AND }(x\in B)$$ $$x\in A^c\iff\text{NOT }(x\in A)$$
So, for example, $A\setminus B=A\cap B^c$, so $$x\in A\setminus B\iff x\in A\cap B^c\iff(x\in A)\text{ AND }(x\in B^c)\iff (x\in A)\text{ AND }(\text{NOT }(x\in B))$$
For completeness, note that XOR is actually inequality on booleans. So, writing boolean equality (equivalence) as $\;\equiv\;$, we have $$x \in A \mathbin\triangle B \;\equiv\; x \in A \;\not\equiv\; x \in B$$ Note that $\;\equiv\;$ and $\;\not\equiv\;$ are both associative, and on top of that they are also mutually associative, so that the above definition does not require any parentheses. – Marnix Klooster Apr 15 '14 at 17:16