set complement

The complement of a set is the set of all elements not in the set.

Formally,
the complement of a set $A$ is written

where $U$ represents the universe and $\setminus$ is the set difference operation.

\documentclass[margin={3cm 0cm}]{standalone}
\usepackage{tikz}
%
\definecolor{fillcolor}{HTML}{8A5CF5}
%
\tikzset{truthy/.style n args={1}{%
    fill=#1,draw,%
    inner sep=0,minimum size=1cm}}
%
\begin{document}
\begin{tikzpicture}
	% not P and not Q
	\begin{scope}[even odd rule]
		\clip (-3,-2) rectangle (3,2) (0,0) circle (1.5);
		\fill[fillcolor] (-3,-2) rectangle (3,2);
	\end{scope}
	%
    \draw (0,0) circle (1.5);
    \node at (0,0) {$A$};
\end{tikzpicture}
\end{document}

Let $A$ and $B$ be two sets in a universe $U$.

De Morgan's laws:

• $(A\cup B{)}^{\complement }=A^{\complement }\cap B^{\complement }$
• $(A\cap B{)}^{\complement }=A^{\complement }\cup B^{\complement }$

Complement laws:

• $A\cup A^{\complement }=U$
• $A\cap A^{\complement }=\varnothing$
• ${\varnothing }^{\complement }=U$
• $U^{\complement }=\varnothing$
• If $A\subseteq B$, then $B^{\complement }\subseteq A^{\complement }$
• ${\left(A^{\complement }\right)}^{\complement }=A$