Question

Prove that for all sets X, Y , if X ∪ Y = X ∩ Y , then X = Y .

Answer 1

We show X = Y by proving that both X ⊆ Y and Y ⊆ X. Assume X ∪ Y = X ∩ Y.

• Prove X ⊆ Y :

Let x ∈ X. Since X ⊆ X ∪ Y, we know x ∈ X ∪ Y. But X ∪ Y = X ∩ Y, which means x ∈ X ∩ Y, and by definition of intersection, x ∈ Y. Therefore X ⊆ Y.

• Prove Y ⊆ X :

This proof is the same, just swap X with Y in the previous one.

Hence X = Y.