Set A is a subset of set B if for every element in A, that element exists in B.

Set A is a proper subset of set B if A is a subset of B, but B is not a subset of A.

So {Fred, Ashraf} is a subset of S because both Fred and Ashraf are in S.

But S is not a subset of {Fred, Ashraf} because Sue is not in {Fred, Ashraf}

Now take it one step further:

Prove that if A is a proper subset of B, then A is a subset of B.

Proof: For A to be a proper subset of B, A must be a subset of B.

Therefore A is a subset of B. QED.