1. Recall that for sets x and Y , the ‘Cartesian product’ X × Y is the set of all pairs (x, y), with x ∈ X and y ∈ Y . [i] How many elements are in Z/3Z × Z/5Z? What’s the difference between this set and Z/5Z × Z/3Z? [ii] If it makes sense to add a and b, and it makes sense to add c and d, then it makes sense to add the pairs (a, c) and (b, d), by the rule (a, c) + (b, d) = (a + b, c + d) For instance, if we take (1, 2) and (2, 1) in Z/3Z × Z/5Z, then (1, 2) + (2, 1) = (3, 3) = (0, 3) Is the set Z/3Z × Z/5Z with this addition a group? Why? [iii] What happens if you add (1, 1) to itself repeatedly (that is, g