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, ���nd 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.
(f) If two integers have opposite parity, then their product is even. (Use formal de���nitions of odd and even numbers in your proof!)