# complement

## 1 Definition

Let $A$ be a subset of $X$. The complement of $A$ in $X$ (denoted $A^{\complement}$ 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

• $(A^{\complement})^{\complement}=A$

• $\emptyset^{\complement}=X$

• $X^{\complement}=\emptyset$

• If $A$ and $B$ are subsets of $X$, then $A\setminus B=A\cap B^{\complement}$, 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

 $\displaystyle\left(\bigcup_{i\in I}A_{i}\right)^{\complement}$ $\displaystyle=$ $\displaystyle\bigcap_{i\in I}A_{i}^{\complement},$ $\displaystyle\left(\bigcap_{i\in I}A_{i}\right)^{\complement}$ $\displaystyle=$ $\displaystyle\bigcup_{i\in I}A_{i}^{\complement}.$
Title complement Complement 2013-03-22 12:18:51 2013-03-22 12:18:51 CWoo (3771) CWoo (3771) 7 CWoo (3771) Definition msc 03E99 DeMorgansLaws