Discreet math homework/MAD 2104 – ONLINE – Spring 2017 Written Assignment 1 This assignment covers material from Module 1 Lesson 1-3

1. Let P, Q, and R be three statements. Determine if the following two statements are logically equivalent: P → (Q∧R) and (∼ Q∨∼ R) →∼ P You may use the following

table to organize your solution: [10 points]

P Q R ∼ P ∼ Q ∼ R (Q∧R) (∼ Q∨∼ R) P → (Q∧R) (∼ 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 P,Q,R,S be four mathematical statements. Suppose P is a false and (R → S) ↔ (P ∧Q) is a true statement, ﬁnd the truth values of R,S. [10 points] (This can be

done without a truth table.)

3. Consider the following statement and its proof. What’s 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 −2·4 + 5 = 4−4 + 5 = 5 is odd.”) (b) Suppose x,y ∈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 deﬁnitions of odd and even numbers in your proof!)

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