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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 indexset. In other words, $I$ can be finite, countable, or uncountable. Then
$\displaystyle\left(\bigcup_{{i\in I}}A_{i}\right)^{\complement}$  $\displaystyle=$  $\displaystyle\bigcap_{{i\in I}}A_{i}^{\complement},$  
$\displaystyle\left(\bigcap_{{i\in I}}A_{i}\right)^{\complement}$  $\displaystyle=$  $\displaystyle\bigcup_{{i\in I}}A_{i}^{\complement}.$ 
Related:
DeMorgansLaws
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