Let the collection i = {a_ alpha: alpha elementof i} be a partition of a nonempty set a and j = {b_ beta: beta elementof j} be a partition of a nonempty set b. we define the following collection of sets w = { a_ alpha times b_ beta: alpha elementof i and beta elementof j}. (a) give a "small" example of this situation using a = b = r and two different partitions i and j, each having exactly two sets. (specify i, j, and w.) (b) does the collection w for your example above partition the plane r times r? (no proof needed for this part, just state your answer.) (c) prove that with i, j and w as defined at the beginning of this problem, w is a partition of a times b.