3. Prove that if n is an integer and n³+5 is odd, then n is even using poa) proof by contrapositionb) proof by contradictionCS Scanned with CamScanner

Answered: 3. Prove that if n is an integer and…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

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3. Prove that if n is an integer and n³+5 is odd, then n is even using po
a) proof by contraposition
b) proof by contradiction
CS Scanned with CamScanner

Expert Solution

Step 1

a) Proof of contraposition: To prove that if n is an integer and n^3 + 5 is odd, then n is even, we will prove the contrapositive, which is that if n is an odd integer, then n^3 + 5 is even.

Assume that n is an odd integer. Then, there exists an integer k such that n = 2k + 1. We can substitute this value of n into n^3 + 5 to get:

n^3 + 5 = (2k + 1)^3 + 5 = 8k^3 + 12k^2 + 6k + 1 + 5 = 8k^3 + 12k^2 + 6k + 6

We can factor out a 2 from the last two terms to get:

n^3 + 5 = 2(4k^3 + 6k^2 + 3k + 3)

Since 4k^3 + 6k^2 + 3k + 3 is an integer, we have shown that n^3 + 5 is even. Therefore, the contrapositive is true, and so is the original statement.

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