Let P be the set of all people and R be a relation defined on P as described in the parts a, b, c, and d below. Determine if R has the reflexive, symmetric, antisymmetric, or transitive properties. Assume x and y are arbitrary members of P. a. x Ry if x is the sister of y. b. x Ry if x and y have the same father. c. x Ry if x and y have the same mother. d. x Ry if x and y have the same paternal grandfather.

7. (Relations) Consider the set A= {1,2, 3, 4, 5}, and a relation R defined by x Ry x2 – y is an integer multiple of 3. (a) List all elements y for which 1Ry is true. (b) Is R reflexive? Justify. (c) Is R symmetric? Justify. (d) Is R antisymmetric? Justify. (e) Is R transitive? Justify.

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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 AXB I (x, y) ER or (x, y) E S} RNS = {(x, y)E AXB| (x, y) ER and (x, y) Es}. Let A = {-2, 2, 3, 5} and B = {2, 3} and define relations R and S from A to B as follows: For every (x, y) E Ax B, x R y - |x| = lyl and xsyex - y is even. Using set-roster notation, state explicitly which ordered pairs are in A x B, R, S, RU S, and RN s. (Enter your answers as comma-separated lists of ordered pairs.) AXB = R = S = RUS = ROS =