PlanetMath: de Morgan's laws for sets (proof)

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de Morgan's laws for sets (proof)

(Proof)

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. We first show that

$\displaystyle \displaystyle \big( \cup_{i\in I} A_i \big)'$

$\displaystyle =$

$\displaystyle \cap_{i\in I} A_i',$

where

denotes the complement of

Let us define $S=\big( \cup_{i\in I} A_i \big)'$ and $T=\cap_{i\in I} A_i'$ . To establish the equality , we shall use a standard argument for proving equalities in set theory. Namely, we show that $S\subset T$ and $T\subset S$ . For the first claim, suppose is an element in . Then $x\notin \cup_{i\in I} A_i$ , so $x\notin A_i$ for any $i\in I$ . Hence $x\in A_i'$ for all $i\in I$ , and $x\in \cap_{i\in I} A_i'=T$ . Conversely, suppose is an element in $T=\cap_{i\in I} A_i'$ . Then $x\in A_i'$ for all $i\in I$ . Hence $x\notin A_i$ for any $i\in I$ , so $x\notin \cup_{i\in I} A_i$ , and $x\in S$ .