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.

Use inductive reasoning to decide whether the statement is true or false. Make an example or a counterexample to supplement your answer.Write your answer in a piece of paper, take a picture of it and upload it here. Include your name and section. The product of an odd counting number and an even counting number is always an even counting number.

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)))

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.

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.

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.