# I used the definitions, now what?

1. Jun 24, 2008

### JinM

1. The problem statement, all variables and given/known data

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?

2. Jun 24, 2008

### 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.

3. Jun 24, 2008

### 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.

4. Jun 25, 2008

### JinM

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