Need help with (𝐴Δ𝐵)×(𝐴∩𝐵)

Question

𝐴 = {∅,{ℕ}} and 𝐵 = {{∅,ℕ},∅}.  Find P((𝐴Δ𝐵)×(𝐴∩𝐵)).

What I have right now is:

𝐴Δ𝐵 = {{ℕ}, {∅,ℕ}}

𝐴∩𝐵 = {∅}

A x B = {({ℕ}, ∅), ({∅,ℕ}, ∅)}

Is that correct?

Answer 1

Your symmetric difference of A and B -- notated as A Δ B -- is correct. To find it we take A ∪ B and subtract its intersection, A ∩ B. In words, it's the elements which are not shared by both sets.

The intersection you've found: A ∩ B, is also correct. To find the intersection of two sets, we look at all of the elements shared by both sets.

When taking the cross-product, A x B, we're looking at the possible ways to form ordered pairs using elements taken from each set. For A x B, we should form all possible pairs (a, b), with values of a taken from A and values of b taken from B. Ordered pairs are typically notated using parentheses. Your solution to the cross-product has the right idea. Based on your answer, you have correctly paired up the elements from each side, to pair {ℕ} and {ℕ, ∅} from A with ∅, from B.

However, the question asked you to take the power-set of this set. The power set P(A) of A is formed by looking at all possible subsets of a set. For instance, the power set of {a, b} is the set { ∅, {a}, {b}, {a, b} }. In order to complete the question, you will need to apply this definition to the cross-product.