Symmetric difference

Definition

Let $A$ and $B$ be sets. The symmetric difference of $A$ and $B$ is the set $$A \mathbin{\triangle} B = (A \setminus B) \cup (B \setminus A).$$ An object $x$ is an element of $A \mathbin{\triangle} B$ when $x$ is an element of exactly one of $A$ and $B$.

Remarks

Another notation is $A \ominus B$. Example: $\{1, 2, 3\} \mathbin{\triangle} \{2, 3, 4\} = \{1, 4\}$. Also $A \mathbin{\triangle} B = (A \cup B) \setminus (A \cap B)$. The power set of a set is a group under symmetric difference, with identity $\varnothing$.

Explanations

L1 human

The symmetric difference is the set version of "exclusive or". It keeps what is in one set or the other, but not what is in both.