Show that (X ∪ Y) ∁ = X∁ ∩ Y∁
First we must show that (X ∪ Y) ∁ ⊆ X∁ ∩ Y∁
Let x ∈ ( X ∪ Y) ∁ . For this to be true, x cannot be in either X or Y so x has to belong to both X∁ and Y∁ . As such, we can be sure that x will be in neither X nor Y. Therefore, (X ∪ Y) ∁ ⊆ X∁ ∩ Y∁
Next we must show that X∁ ∩ Y∁ ⊆ (X ∪ Y) ∁
Let x ∈ X∁ ∩ Y∁ . This means that x is not in X and not in Y. Since this is assumed to be true, x would have to belong to neither X nor Y as that would ensure that x would not be in both the complement of X and the complment of Y so x ∈ ( X ∪ Y) ∁ .
Since (X ∪ Y) ∁ ⊆ X∁ ∩ Y∁ and X∁ ∩ Y∁ ⊆ (X ∪ Y) ∁ , we have shown that (X ∪ Y) ∁ = X∁ ∩ Y∁