The symmetric difference

The symmetric difference contains the elements that belong to only one of the two sets, excluding those in common. It is the logical “aut”: “either one or the other, but not both”.

Definition — Symmetric difference

The symmetric difference of $A$ and $B$ is: $A\mathbin{\triangle} B = (A\cup B)\setminus(A\cap B) = (A\setminus B)\cup(B\setminus A).$ It contains the elements that belong to only one of the two sets.

The symmetric difference $A\triangle B$: the two outer “slices”, without the common region.

Remark — Exclusive OR

In logic:

• $\cup$ corresponds to the vel (inclusive OR, “either… or… or both”).
• $\triangle$ corresponds to the aut (exclusive OR, “either… or… but not both”).

Links

Topics: Set theory
Concepts: Venn diagram · Symmetric difference · Exclusive disjunction