If S is subset of S', then lin(S) is subset of lin(S')

Question

I have this problem on my linear algebra course, which I don't know how to start solving. The problem follows:

Let V be a vector space and S, S' ⊆ V. Prove right or wrong that S ⊆ S' implies⟹ lin(S) ⊆ lin(S').

The further question is: Does the answer change when examining proper subsets?

That is, does S ⊂ S' imply⟹ lin(S) ⊂ lin(S')?