de Morgan’s laws for sets (proof)
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 .
|Title||de Morgan’s laws for sets (proof)|
|Date of creation||2013-03-22 13:32:16|
|Last modified on||2013-03-22 13:32:16|
|Last modified by||mathcam (2727)|