CSI 2101 (W19) - MIDTERM TEST SOLUTION (For questions 1 and 2 you do not need to show your work, it will not be marked, even if you do show it) Question 1. [4 points] Suppose P(x, y) is a predicate and the universe for the variables x and y is {1, 2, 3}. Suppose P(1, 3), P(2, 1), P(2, 2), P(2, 3), P(3, 1), P(3, 2) are true, and P(x, y) is false otherwise. Determine the truth value of the statements: ¬xy (P(x, y) ˄ ¬P(y, x)) FALSE It is equivalent to xy ¬ (P(x, y) ˄ ¬P(y, x)) = xy (¬ P(x, y) ˅ P(y, x)) And this is not true when x= 2 and y=1 yx (P(x, y) → P(y, x)) TRUE It is equivalent to yx (¬ (P(x, y) ˅ P(y, x)). Easy to check for y=1, 2 and 3 yx (x y ˄ P(x, y)) FALSE For y=1 the only x where x y is x=1. But P(1, 1) is false. xy (x = y ˄ P(x, y)) x=2 and y=2 satisfy the formula. TRUE Question 2. [2 points] (Circle the correct answer) The formula 5 “ y x M(x, y)“ corresponds to 'Everybody has a mother'? Question 3 [6 marks] Determine whether the following argument is valid (write clearly your proof): p→r q→r ¬ (p ˅ q) ⸫ ¬r Not valid, enough to consider the counter example p=F, q=F and r=T. All hypothesis are true but not the conclusion. Question 4 [6 marks] Prove or disprove the following: If p and q are prime numbers, then p + q is not a prime number. Disprove: enough to consider the counter example p=2 and q=3, both are primes but p+q=5 is also prime. Question 5. [7 points] Prove the following theorem: Let n be an integer. If n2 – 2n +3 is even, then n is odd. (Indicate clearly which type of proof you are using) Proof by contrapositive: Suppose n is even thus n= 2k. Therefore n2 – 2n +3 = (2k)2 – 2(2k) +3 = 2(2k2 – 2k +1) +1. Thus n2 – 2n +3 is odd. 2