Home Twitter Tumblr blog

Maths Proof

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