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?
I have the same exact question.