Set Operations — Difference, Symmetric Difference and Complement

Three more operations round out the standard set-operation toolkit.

The difference of $A$ and $B$, written $A - B$, is the set of elements that belong to $A$ but not to $B$:

Note carefully that $A - B$ and $B - A$ are generally different sets — subtraction of sets, like subtraction of numbers, is not commutative.

The symmetric difference of $A$ and $B$, written $A \triangle B$, is the set of elements belonging to exactly one of $A$ and $B$, but not both — everything in either set except what they share:

For example, with $A = \{1,2,3,4,5\}$ and $B = \{4,5,6,7,8\}$:

Notice $A \triangle B$ is simply $A - B$ and $B - A$ combined — it deliberately leaves out the shared elements $4$ and $5$.

The complement of a set $A$, written $A'$, is defined only relative to a chosen universal set $U$ (the "everything under discussion" set from the previous chapter) — it is everything in $U$ that is not in $A$:

For instance, if $U = \{1, 2, \dots, 10\}$ and $A = \{1,2,3,4,5\}$, then $A' = \{6,7,8,9,10\}$. …