Exercise 578Logic, sets and proofs › Sets and mapsL1Added 3 Nov 2026

Symmetric difference

#symmetric difference#XOR#sets

Statement

Show that $A\triangle B=(A\cup B)\setminus(A\cap B)=(A\setminus B)\cup(B\setminus A)$ .

Hint

Element-wise proof.

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Definition. $A\triangle B=\{x:x\in A\text{ or }x\in B\}$ .

$(A\setminus B)\cup(B\setminus A)$ . $x$ is in this set iff $x$ is in exactly one of $A$ or $B$ .

$(A\cup B)\setminus(A\cap B)$ . $x$ is in this set iff $x$ is in $A\cup B$ but not in $A\cap B$ , i.e. exactly one of $A,B$ .

Conclusion. The three expressions are equal.