hmmm

well go ahead and assume that for a1 and a2 in A j(a1) = j(a2).

By the definition of your function

j(a1) = (a1, b1) for some a1 in A and b1 in B and

j(a2) = (a2, b2) for some a2 in A and b2 in B.

Since j(a1) = j(a2) this implies that (a1, b1) = (a2, b2). But this is only true if a1 = a2 and b1 = b2.