Let 𝐴 and 𝐵 be sets. Using set builder notation, show that:a) 𝐴 ∩ ∅ = ∅.b) 𝐴 ∪ 𝐴 = 𝑈.c) 𝐴 ∩ (𝐵 − 𝐴) = ∅

a) 𝐴 ∩ ∅ = ∅ The intersection of any set A with the empty set is the empty set. This is because the intersection of two sets is the set of elements that are common to both sets. Since the empty set has no elements, it has no elements in common with any set A. In set builder notation, this can be written as: 𝐴 ∩ ∅ = {x | x ∈ A and x ∈ ∅} = ∅

b) 𝐴 ∪ 𝐴 = 𝑈 The union of any set A with itself is just the set A. This is because the union of two sets is the set of elements that are in either set. Since all elements of A are in A, the union of A with itself is just A. However, the symbol 𝑈 is typically used to represent the universal set, which contains all elements under consideration. So unless A is the universal set, 𝐴 ∪ 𝐴 = 𝑈 is not correct. It should be 𝐴 ∪ 𝐴 = 𝐴. In set builder notation, this can be written as: 𝐴 ∪ 𝐴 = {x | x ∈ A or x ∈ A} = A

c) 𝐴 ∩ (𝐵 − 𝐴) = ∅ The set B - A is the set of elements that are in B but not in A. Therefore, the intersection of A with (B - A) is the empty set, because there are no elements that are in both A and in B but not in A. In set builder notation, this can be written as: 𝐴 ∩ (𝐵 − 𝐴) = {x | x ∈ A and x ∈ (B - A)} = ∅