Complement

Fix a set $U$ (the “universe”). For $A \subseteq U$ (see subset ), the complement of $A$ in $U$ is

where $\setminus$ is set difference .

Complements exchange unions and intersections via De Morgan’s laws (relative to $U$).

Examples:

• If $U=\{1,2,3,4,5\}$ and $A=\{2,5\}$, then $A^c=\{1,3,4\}$.
• In $U=\mathbb{R}$, the complement of $(0,1)$ is $(-\infty,0]\cup[1,\infty)$.