Assignment 1 Pair Work Due 24 November 2014 before 5pm Instructions: Answers must be typed and submit through Morpheus and also hardcopy. You can use the MatType editor at http://www.dessci.com/en/products/mathtype/ to type the equations/symbols in your answers. The softcopy must be submitted in Microsoft word file format to Dr Lee’s Morpheus. Only one member from a pair is needed to submit. The hardcopy is to be submitted to your respective lecturer. Question 1 Let A = {a, b, c, d, e} and S, T, U, and V relations on A where S = {(a, a), (a, b), (b, c), (b, d), (c, e), (e, d), (c, a)} T = {(a, a), (b, a), (b, c), (b, d), (e, e), (d, e), (c, b)} U = {(a, b), (a, a), (b, c), (b, b), (e, e), (b, a), (c, b), (c, c), (d, d), (a, c), (c, a)} V = {(a, b), (b, c), (b, b), (e, e), (b, a), (c, b), (d, d), (a, c), (c, a)} Find the represenation matrices for S, T, U, and V. Then uses these matrices to determine which of the relations are symmetric, reflexive, transitive, or/and antisymmetric. Question 2 Use a proof by contradiction to prove that if x 2  x  2  0 then x  0 Question 3 Determine if the equations below are one-to-one or onto or both. Explain your answer clearly. a) x = 7 b) y = 5 c) y2 = x2 + 4 d) y = x2 + 4

Question 4 Given A = {2, 3, 4}, B={b, c}. a) Find A x B b) Find A x  c) Find B x B

Question 5 a) Determine whether the relation R on the set Z is having reflexive, symmetric, and/or transitive property. a) (a, b)  R if a + b = 5. b) (a, b)  R if a + b = 0.

Question 6 Show that if A and B are sets, then a) A  B  A  B . b) A  B  A . Use element wise method to show your proof.