Definition

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

The Venn diagram below illustrates the complement of in red.

$\begin{pspicture}(0,0)(8,6) \pspolygon[fillstyle=vlines,hatchcolor=red,hatchwidt... ...etminus A$} \rput(8.5,5.75){$X$} \rput(0,0){$.$} \rput(8,6){$.$} \end{pspicture}$

Properties

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

de Morgan's laws

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

$\displaystyle \left( \bigcup_{i\in I} A_i \right)^\complement$	$\displaystyle =$	$\displaystyle \bigcap_{i\in I} A_i^\complement,$
$\displaystyle \left( \bigcap_{i\in I} A_i \right)^\complement$	$\displaystyle =$	$\displaystyle \bigcup_{i\in I} A_i^\complement.$

"complement" is owned by CWoo. [ full author list (2) | owner history (1) ]

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