Question Concerning Proof of Equality of Sets...?

Question

Okay, if have a problem where I have to prove the equality of the following sets:

A intersection ( B union C ) = ( A intersection B ) union ( A intersection C )

If D is a subset of A and D is a subset of B, then D is a subset of ( A intersection B )

If A is a subset of D and B is a subset of D, then ( A union B ) is a subset of D

Thanks in advance! I'll continue to work on this problem for a while before checking back, so feel free to take your time, I'll check back in 1.30 hrs

Answer 1

For the first one, I would try to prove the biconditional statement: if (right side) then (left side), AND if (left side) then (right side). e.g. Assume that an element is part of the right hand set. Then show that it must be in the left hand set. Likewise, assume that an element is in the left hand set, show that it is part of the right hand set.

The second and third are not equivalence proofs--they go only one way. I'd prove the second by assuming that D is NOT an element of A int B and then forcing a contradiction using the definitions. Likewise the third should succumb to the indirect method.