Set Symmetric Differences

Why

We want to consider the non-overlapping elements of a pair of sets.

Definition

In other words, we want to consider the set of elements which is one or the other but not in both. The symmetric difference (or Boolean sum) of a set with another set is the union of the difference between the latter set and the former set and the difference between the former and the latter.

Notation

Let $A$ and $B$ denote sets. We denote the symmetric difference by $A + B$, so that

\[ A + B = (A - B) \cup (B - A) \]

Properties

Here are some immediate properties of symmetric differences.¹

$A + B = B + A$.

$(A + B) + C = A + (B + C)$.

$(A + \varnothing) = A$

$(A + A) = \varnothing$

Future editions will have more detailed (but obvious) hypotheses stated. ↩︎