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Let be a set with subsets
for , where is an arbitrary index-set. In other words, can be finite, countable, or uncountable. We first show that
where denotes the complement of .
Let us define
and
. To establish the equality , we shall use a standard argument for proving equalities in set theory. Namely, we show that
and
. For the first claim, suppose is an element in . Then
, so
for any . Hence for all , and
. Conversely, suppose is an element in
. Then for all . Hence
for any , so
, and .
The second claim,
follows by applying the first claim to the sets .
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