1. Let the statement be “If n is not an odd integer then square of n is not odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “(square of n) is not odd.” For direct proof we should prove _________a) ∀nP ((n) → Q(n))b) ∃ nP ((n) → Q(n))c) ∀n~(P ((n)) → Q(n))d) ∀nP ((n) → ~(Q(n)))

3. Let the statement be “If n is not an odd integer then sum of n with some not odd number will not be odd.”, then if P(n) is “n is an not an odd integer” and Q(n) is “sum of n with some not odd number will not be odd.” A proof by contraposition will be ________a) ∀nP ((n) → Q(n))b) ∃ nP ((n) → Q(n))c) ∀n~(P ((n)) → Q(n))d) ∀n(~Q ((n)) → ~(P(n)))

6. Use a direct proof to show that the sum of two odd integers is even.7. Use a direct proof to show that the sum of two even integers is even Show that if n is an integer andn3 + 5 is odd, then n is even usinga) a proof by contraposition.b) a proof by contradiction.8. Prove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd.9. Find a counterexample to the statement that every positive integer can be written as the sum of thesquares of three integers.

Use inductive reasoning to decide whether each statement is true. a.   The sum of two odd counting numbers is always an odd counting number.  b.   The product of two odd counting numbers is always an odd counting number.

Given:(1) If n is an even number, then 5n + 2 is always an even number.(2) If n is an integer larger than 1, then 2^n − 1 is always a prime number.Select one of the following choices:Question 16Answera.(1) and (2) are True.b.(1) is True and (2) is False.c.(1) and (2) are False.d.(1) is False and (2) is True.

For each of the following statements, either prove that the statement is true, or finda counterexample to show that the statement is false and explain your reasoning.(a) For each integer n, n is even if and only if 6n + 4 is even.(b) For all integers ℓ, m, n, if ℓ + m is odd and ℓ + n is even, then m + n is odd.