Disprove the following statement by giving a counterexample. For every integer n, if n is even then n2 + 1 is prime. Counterexample: Consider the ordered pair (n, n2 + 1) The values in the ordered pair show that the given statement is false because (choose one) n is even and n2 + 1 is prime. n is even and n2 + 1 is not prime. n is not even and n2 + 1 is prime. n is not even and n2 + 1 is not prime.

In each of the following statements, n represents a positive integer. One of the statements is true and the other two are false.  If n is even, then n is not prime  if n is a square number, then n is not prime  if n is a prime number and n ≥ 10, then n+2 is prime or n+4 is prime.  A. What two statements are false and in each case give a counter example to show that it is false  B. Prove the true statement

Use the PMI to prove that 3 - 1 is even for all n E N. Proof. Base case: Since 3¹ - 1 = 2, which is even. Thus the statement is true for n = [Select] Inductive step: Assume that there is a natural number n such that 3 - 1 is even. Then 3-1 = [Select] for some integer k. Then 3" [Select] ✓. On both sides, first multiply by 3, then subtract 1, and simplify to get 3+1 -1 = 2( [Select] which is an [Select] integer. Hence, by the PMI, 3 - 1 is even for all n E N. )

3. Prove that if a is an odd number and b is an integer then b² # a² +9. Hint: You will need to use the lemma "If n² is even then n is even" to complete this proof.

4. Explain why the following statements are true or False. (a) There is an even number that is a multiple of 3. (b) All prime numbers are odd. (c) Vm,neN, m •nzm+ n

3. Which of the following operations is BOTH associative and commutative on positive integers C. a*b = 2- (Z+)? min(a,b) D. a*b = a-b A. a*b = 2b + a B. a*b = max(a,b) 3

Prove that if an integer m has the same parity as an integer n, 5m + 7n is even. Discrete math