complement

1 Definition

Let $A$ be a subset of $X$. The complement of $A$ in $X$ (denoted $A^{\mathrm{\complement }}$ when the larger set $X$ is clear from context) is the set difference $X\setminus A$.

The Venn diagram below illustrates the complement of $A$ in red.

2 Properties

• ${(A^{\mathrm{\complement }})}^{\mathrm{\complement }}=A$

• ${\mathrm{\varnothing }}^{\mathrm{\complement }}=X$

• $X^{\mathrm{\complement }}=\mathrm{\varnothing }$

• If $A$ and $B$ are subsets of $X$, then$A\setminus B=A\cap B^{\mathrm{\complement }}$, where the complement is taken in $X$.

3 de Morgan’s laws

Let $X$ be a set with subsets $A_i\subset X$ for $i\in I$, where$I$ is an arbitrary index-set. In other words, $I$ can be finite,countable, or uncountable. Then

${\left({\displaystyle \bigcup _{i\in I}}{A}_i\right)}^{\mathrm{\complement }}$$=$${\displaystyle \bigcap _{i\in I}}{A}_i^{\mathrm{\complement }},$

${\left({\displaystyle \bigcap _{i\in I}}{A}_i\right)}^{\mathrm{\complement }}$$=$${\displaystyle \bigcup _{i\in I}}{A}_i^{\mathrm{\complement }}.$