Please help with logical equivalences!!!

Question

Let  and  be any given sets. Use set builder notation and logical equivalences to show that       .

Answer 1

Let x is any element from left side set x∈(AUB)'. Then x ∉ (AUB) so x ∉ A and x ∉ B, So x ∈ A' and x ∈ B' which means that x ∈ A' ∩ B'.

Now let y is any element from right side, so y ∈ A' ∩ B'. So y ∈ A' and y ∈ B'; so y ∉ A and y ∉ B then

y ∉ A U B. So y ∈ (A U B)'.

This logic prove that (A U B)' = A' ∩ B'.