I'm taking Topology next semester and, to prepare, I'm learning a little set theory. I had bought a Topology book from Dover Publications a couple years ago and am now starting to do some problems in it. The very first section is on Set Theory. Here's the first couple problems:

1. S ⊂ T, then T - (T - S) = S.

Proof:

Let x ∈ (S ⊂ T). Therefore, x ∉ (T – S). If x ∉ (T – S), then x ∈ T – (T – S). And x ∈ (S ⊂ T). Therefore, S ⊂ (T - (T - S))

Let x ∈ (T - (T - S)). This implies that x ∉ (T – S). Which implies that x ∈ S. Therefore, T – (T – S) ⊂ S and T - (T - S) = S.

2. S ⊂ T iff S ∩ T = S.

Proof:

If S ⊂ T, every x ∈ S is in T. That implies that every x ∈ S is in S ∩ T. This implies that S ∩ T = S.

Let x ∈ S. If S ∩ T = S, then x ∈ (S ∩ T). This implies that S is contained in T and S ⊂ T.

I could really use some direction/correction.