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)))

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)))

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.

Statement A (Assertion): a, b, c are in AP if and only if 2b = a + c. Statement B (Reason): The sum of first n odd natural numbers is n². e. Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion(A). f. Both assertion (A) and reason (R) are true and reason (R) is not the correct explanation of assertion (A). g. Assertion (A) is true but reason (R) is false. h. Assertion (A) is false but reason (R) is true.

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.