Answer
for both answers see the step by step sloution...
Work Step by Step
This exercise asks for a proof of one of De Morgan's laws for sets. The primary way to show that two sets are equal is to show that each is a subset of the other. In other words, to show that X = Y, we must show that whenever x ϵ X, it follows that x ϵ Y, and that whenever x ϵ Y, it follows that x ϵ X.
a) Suppose x ϵ (AU B)ᶜ.
Then x ∉ (AU B) . which means that x is in neither A nor B. In other words, x ∉ A and x ∉ B. This is equivalent to saying that x ϵ Aᶜ and x ϵ Bᶜ. Therefore x ϵ Aᶜ Ո Bᶜ as desired. Conversely, if x ϵ Aᶜ Ո Bᶜ then x ϵ Aᶜ and x ϵ Bᶜ. This means x ∉ A and x ∉ B, so x cannot be in the union of A and B . Since x ∉ (A U B), we
conclude that x ϵ (A U B)ᶜ, as desired.
b)The following membership table gives the desired quality, since coulmns four and seven are identical.
A B (AUB) (AUB)ᶜ Aᶜ Bᶜ AᶜՈBᶜ
1 1 1 0 0 0 0
1 0 1 0 0 1 0
0 1 1 0 1 0 0
0 0 0 1 1 1 1
