Prove that .

* Proof. * First, let

be any element in

. By definition of complement, this means that

and

. Since

we have either

or

which implies, coupled with the fact that

is in

, means

or

, respectively. Since

is in at least one of these,

is in the union

. Therefore,

.

For the reverse inclusion, let

be an arbitrary element of

. Then, either

or

.

If

, then

and

; hence,

. Therefore,

* is * in

. On the other hand, if

, then

and

; hence,

. This again implies

is in

. Therefore,

.

Hence,

.∎

*Related*

sorry, i cant understand Since x \notin (B \cap C) we have either x \notin B or x \notin C which

why this happens?