2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a, b) : a € A and be B}. This can be extended naturally to a finite number of sets A1, A2,..., A: A1 x A2 x -..x Ak = {(a1, a2, ...,ak) : a; € Aj for 1< j< k}. %3D This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by II x X, = {s :1¬UX; : f6) e X, for all i e I iel That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X; for any i € I. Let {G;}ie1 be a family of groups indexed by the set I. Let e; be the identity element of Gi, for all i e I. Consider the Cartesian product G = IIie, Gi- %3D a. Show that G is a group under multiplication given by fg(i) = f(i)g(i) for all e I. We call G the direct %3D product of the family {G;}ie1-

The Cartesian product of finite and nonempty sets can be illustrated by the intersections of a crossing-line pattern, as shown to the right. {f, g. h} x {x, y} (a) Explain why the number of intersection points in the crossing-line pattern for AxB is a•b, where a = n(A) and b=n(B). (a) Each of the lines coming from set A intersects each of the lines coming from set B