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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.
- $(A^{\complement})^\complement=A$
- $\emptyset^\complement = X$
- $X^\complement = \emptyset$
- If $A$ and $B$ are subsets of $X$ , then $A\setminus B = A\cap B^\complement$ , where the complement is taken in $X$ .
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
\begin{eqnarray*} \left( \bigcup_{i\in I} A_i \right)^\complement &=& \bigcap_{i\in I} A_i^\complement, \\ \left( \bigcap_{i\in I} A_i \right)^\complement &=& \bigcup_{i\in I} A_i^\complement. \end{eqnarray*}
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