Theorem
Let $ A $ be a set in S. Then
A ∩ S = S
where S is the universal set and A is a set in S
Proof
Let x ∈ S, where S is the universal set.
First we show that A ∩ S ⊂ A.
We know this is true because the set resulting from the union of two sets is a subset of both of the sets (proof).
Next, we want to show that A ⊂ A ∩ S.
Let x ∈ A. Then x ∈ S such that x ∈ A. Therefore, x ∈ (A ∩ S) ⇒ x ∈ (A ∩ S) ⇒ A ⊂ A ∩ S.
Since A ∩ S ⊂ A and A ⊂ A ∩ S, we have that A ∩ S = A.
$ \blacksquare $
