Complement

Fix a universe set $U$ (an ambient set containing all sets under discussion). The complement of $A\subseteq U$ (relative to $U$) is

Complements are fundamental in topology and measure theory because “closed” is defined as “complement of open,” and many set identities are most naturally expressed using complements.

Examples:

• If $U=\mathbb{R}$ and $A=[0,1]$, then $A^c = (-\infty,0)\cup(1,\infty)$.
• If $U=\{1,2,3\}$ and $A=\{1,3\}$, then $A^c=\{2\}$.
• If $A=\varnothing$, then $A^c=U$; if $A=U$, then $A^c=\varnothing$.