“Discreet math homework/MAD 2104 ONLINE Spring 2017 Written Assignment 1 This assignment covers material from Module 1 Lesson 1-31. Let P Q and R be three statements. Determine if the following two statements are logically equivalent: P (QR) and ( Q R) P You may use the following table to organize your solution: [10 points]
P Q R P Q R (QR) ( Q R) P (QR) ( Q R) P T T T T T F T F T T F F F T T F T F F F T F F F
2. Let PQRS be four mathematical statements. Suppose P is a false and (R S) (P Q) is a true statement nd the truth values of RS. [10 points] (This can be done without a truth table.)
3. Consider the following statement and its proof. Whats wrong with this proof [10 points] Let x and y be two positive numbers. If x y then x y. Proof : Suppose x y. Taking the square of both sides we get x y which is true. Therefore x y. 4. Negate the following statements (a) [5 points] The square of every real number is non-negative. (b) [5 points] If x is a rational number then x is not a prime number. (c) [5 points] The number x is even or the number y is even. (d) [5 points] For every prime number p there exists another prime number q with q > p.
5. Prove the following statements using direct proof.[10 points each] (a) If x is an even integer then x2 6x + 5 is odd. (Hint: The following is NOT a proof of this statement: Let x = 2 then x2 6x + 5 = 22 24 + 5 = 44 + 5 = 5 is odd.) (b) Suppose xy R. If x > y then y3 + yx2 > x3 + xy2. (c) If n is an odd integer then n2 1 is a multiple of 8. (d) Suppose that a b c are integers. Prove that if a2|b and b3|c then a6|c. (e) If d is an integer with d > 2 then the equation x2 + 3x + d = 0 has no real solution.
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(f) If two integers have opposite parity then their product is even. (Use formal denitions of odd and even numbers in your proof!)
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