Prove that for any sets A and B that A n B is a subset of A

Solution

Suppose A and B are any sets .

There arise two cases : and suppose x is any element of

Case1. Suppose A and B are disjoint then A B = .

An empty set is subset to every set so that A B = A U B .

Thus we have ( A B ) (A U B) ---------------   ( 1 )

Case 2 : Suppose A and B are not disjoint then A B .

So that there exists atleast one element say x A B.

Then x A and x B also x A U B. which implies that x (A B ) (A U B) -------- ( 2 )

Thus in either two cases it is clear that (A B ) (A U B)