Discrete

Problem 2 (25 points) 1. (15 points) Using a proof by contradiction, show the following. In each case write what you would suppose and what you would need to prove the statements. a. Prove that if a is rational and b is irrational, then their sum a+b must be irrational. b. Show that there cannot exist integers m and n such that m2 - 4n-2. c. Suppose n is a composite number. Then n has a prime divisor which is less than Vn. Note: When the original statement has the form p -> q, or equivalently p v q, then in a proof by contradiction we must assume the negation of the original statement. In this case the negation becomes pAq 2. (5 points) Using a proof by contrapositive, show that if the product of two real numbers x and y is 3. (5 points) Prove using 3 different methods that if n is an even integer, then n+7 is odd. 4. 5 points BONUS Consider the function P(x)-ax +bx+x + d for which you are told that P(O)-1 greater than 16, then at least one of the numbers is greater than 4. and P(r+1)-Px)-6x. a. Find the coefficients a, b, c, and d b. Compute the sum S- 12+22+3*+ + 1000* using what you know about P) only