Question

Let A = {\bullet ,\square, \bigotimes} and B = {\square,\ominus, \bullet}.

Answer 1

(a) With A = {•, □, ⊗} and B = {□, ⊖, •}, we have

A × B = {(•, □), (•, ⊖), (•, •), (□, □), (□, ⊖), (□, •), (⊗, □), (⊗, ⊖), (⊗, •)}

and

B × A = {(□, •), (□, □), (□, ⊗), (⊖, •), (⊖, □), (⊖, ⊗), (•, •), (•, □), (•, ⊗)}

(b) The intersection of the two sets above is

(A × B) ∩ (B × A) = {(•, •), (•, □), (□, •), (□, □)}

Not sure what µ is supposed to represent, but I suppose you meant to again write × as in the Cartesian product. By definition, for any two sets A and B, we have

A × B = {(a, b) | a ∈ A and b ∈ B}

Then

(A × B) ∩ (B × A) = {(a, b) | a ∈ A ∩ B and b ∈ A ∩ B}

In the product found above, notice that • and □ are both elements of A and B, while ⊗ and ⊖ are exclusive to either set.