Suppose A times B subset B times C for some sets A B and C P

Suppose A times B subset B times C for some sets A, B, and C. Prove that A subset C.

Solution

From the given information we conclude that

For any (a,b) in AxB ,(a,b) is in BxC

BUt a is in A and (a,b) is in BxC means a is in B

Hence, A is subset of B

Now b is arbitrary and hence belongs to C because (a,b) belongs to BxC

Hence, B is subset of C

Hence, A subset of C