Suppose that a and b are integers. (a) (5 pts) Use a proof by contradiction to...

Suppose that a and b are integers.

(a) (5 pts) Use a proof by contradiction to prove the statement ”If a − b is even, then a and b have the same parity (that is, they are both even or both odd).” Carefully show your algebra.

(b) (5 pts) Show that any odd integer cubed is also an odd integer. Carefully show your algebra.

(c) (5 pts) Use your results from parts (a-c) in a direct proof to show that ”If a − b is even, then a 3 − b 3 is even.”

(5) (a) (2.5 pts) In your own words what is a base case and why is it important?

(b) (2.5 pts) In your own words what is an inductive hypothesis and how is it used?

(c) (10 pts) Prove that P(n) : 1+24+34+44+· · ·+n 4 = n 30 (n+1)(2n+1)(3n 2+3n−1), ∀n ≥

1. (i) What is P(1)? (ii) Show that P(1) is true. (iii) What do you need to prove in the inductive step? (iv) Complete the inductive step, identifying where you use the inductive hypothesis. (v) Explain why these steps show that this formula is true whenever n ≥ 1.

Expert Solution

d) and e)

As we know, the method called induction requires two cases to be proved. The first case, called the base case (or, sometimes, the basis), proves that the property holds for the number 0. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1. These two steps establish the property P(n) for every natural number n = 0, 1, 2, 3, ... The base step need not begin with zero. Often it begins with the number one, and it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.

Thank you!

venereology answered 2 hours ago

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Suppose that a and b are integers. (a) (5 pts) Use a proof by contradiction to...

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