I used the definitions, now what?

[JinM]

Homework Statement

Prove:

1. $$R \cap ( S \cup T ) = (R \cap S) \cup (R \cap T)$$

2. $$S \cap ( S \cup T ) = S$$

2. The attempt at a solution

I suppose this is all about using the definitions, and I eventually get down to this:

For (1), the LHS is down to x e R and (x e S or x e T), while the RHS is (x e R and x e S) or (x e R and x e T). There's one small leap here, I know. How do I show these two are equivalent?

For (2), I should show that (x e S) and (x e S or x e T) is equivalent to (x e S). What logical conclusion am I missing here?

 

[Dick]

If nothing else, use a truth table. x is either in or is not in R, S and T. That leaves you eight cases. Four in the second one. Is the logic true in all cases? With some moderate cleverness you don't even have to check all eight. Sure, it's just logic.

 

[uman]

for (2), if x is in the lhs then x is in S and (S or T). So x is in the rhs.
If x is in the rhs, x is in S and so x is in (S or T). So x is in S and (S or T), so x is in the lhs.

Thus the lhs is a subset of the rhs and the rhs a subset of the lhs, so the two sets are equal.

 

[JinM]

Thanks guys -- that got me on the right track.