(a) With <em>A</em> = {•, □, ⊗} and <em>B</em> = {□, ⊖, •}, we have
<em>A</em> × <em>B</em> = {(•, □), (•, ⊖), (•, •), (□, □), (□, ⊖), (□, •), (⊗, □), (⊗, ⊖), (⊗, •)}
and
<em>B</em> × <em>A</em> = {(□, •), (□, □), (□, ⊗), (⊖, •), (⊖, □), (⊖, ⊗), (•, •), (•, □), (•, ⊗)}
(b) The intersection of the two sets above is
(<em>A</em> × <em>B</em>) ∩ (<em>B</em> × <em>A</em>) = {(•, •), (•, □), (□, •), (□, □)}
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 <em>A</em> and <em>B</em>, we have
<em>A</em> × <em>B</em> = {(<em>a</em>, <em>b</em>) | <em>a</em> ∈ <em>A</em> and <em>b</em> ∈ <em>B</em>}
Then
(<em>A</em> × <em>B</em>) ∩ (<em>B</em> × <em>A</em>) = {(<em>a</em>, <em>b</em>) | <em>a</em> ∈ <em>A</em> ∩ <em>B</em> and <em>b</em> ∈ <em>A</em> ∩ <em>B</em>}
In the product found above, notice that • and □ are both elements of <em>A</em> and <em>B</em>, while ⊗ and ⊖ are exclusive to either set.
