Answer Happy

Justify and prove everything you state, unless proved in class already; regardless of how a question is worded. You are not meant to solve the problems by any means necessary; solve them using the material, techniques and machinery seen in class. So as to exhibit the familiarity and comfort you should now have with them. Peer discussion is encouraged, however you must write up your own work. If an argument you've used in your solutions is not your own, you will be penalised if you do not cite your source. WHEN WRITING YOUR ANSWERS, PLEASE PAY PARTICULAR ATTENTION TO THE SOUNDNESS AND COMPLETENESS OF YOUR ARGUMENTS; AND ALSO TO THE STRUCTURE, CLARITY (AND LEGIBILITY, IF A WRITTEN SUBMISSION) OF YOUR WRITING. Problem A.1. Negate the following statements. (Given a statement P, just saying "not P" is not enough. Whenever possible, state your negated statements in positive terms. For example, negation of odd should be written even and not "not odd".) (a) If m is even and n is odd, then m+n is odd. (b) There exists integers a and b such that both ab <0 and a + b > 0 are satisfied. (c) For all real numbers x and y, x y implies that x² + y² > 0. (d) For every integer m, there exists an integer n such that m+n is a cube Problem A.2. Let P, Q, R be statements. Prove the following logical equivalences. (You can use truth tables, or you can use logical equivalence laws) (a) P (QVR) Q (PR). (b) (PAQ) R (PA-R)= -Q. Problem A.3. Show that 7x-13 is odd if and only if x is even. Problem A.4. (a) Show that for a = 1, 2, 3, 4, we have a = 1 mod 5 by direct computation. (b) For all integers a such that 5 a, show that a = 1 mod 5. that is, a third-power of some integer