Given that (A ∩ C) ⊆ (B ∩ C) (A ∩ C) ⊆ (B ∩ C) show that A ⊆ B.

To show that A ⊆ B, we need to show that every element in A is also in B. We can use the given information to show this.

Assume that there exists an element x in A such that x is not in B. We will use this assumption to arrive at a contradiction.

Since x is in A and (A ∩ C) ⊆ (B ∩ C), we know that x is also in (A ∩ C). This implies that x is in C.

Since x is not in B but x is in C, we know that x is not in (B ∩ C).

However, we also know that (A ∩ C) ⊆ (B ∩ C). This means that every element in (A ∩ C) is also in (B ∩ C).

Since x is in (A ∩ C) but not in (B ∩ C), we have arrived at a contradiction. Therefore, our initial assumption that there exists an element x in A such that x is not in B must be false.

This implies that every element in A is also in B. Thus, we have shown that A ⊆ B.