Problem 3. Suppose that there is a group of n people, where n 2 2. Prove that there are at least two people in the group with the same number of friends. Some notes: every two people are either friends or are not, if a is a friend of b then b is a friend of a, and no person is their own friend - in other words, friendship is a symmetric, irreflexive relation on this set of n peo- ple. (Hints: use the pigeonhole principle, start by explaining why it's not possible for one per- son to have 0 friends and another person to have n-1 friends.)

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This question refers to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from a set A to a set B, RUS= {(x, y) E A × B| (x, y) ERor (x, y) E S} RO S= {(x, y) E A x B | (x, y) ERand (x, y) E S}. Let A = {-4, 4, 7, 9} and B = {4, 7}, and define relations R and S from A to B as follows. For every (x, y) E Ax B, x R y x| = [y] and x Sy x - y is even. Which ordered pairs are in A x B, R, S, R U S, and RN S? (Use set-roster notation to enter your answers.) Ax B = R = S = RUS = ROS =

A={1,2,3,4,5,6} and consider the following 3 subsets of A. A = {1,3,5}; A2 = Let {2, 4, 6}; A3 = {3, 6}.