12. Consider the following (true) claim and two possible proofs. Claim. If n is a positive integer, then n² + 3n+2 is not a prime number. Proof 1. Note that n² + 3n+2= (n + 2)(n+1). Since n> 1, we have n + 1>1 and n+2 > 1, so n²+3n+2 has at least two divisors other than 1 and is therefore not prime. Proof 2. If n is even then we can write n = 2k for some positive integer k. Then n² + 3n+2 = 4k² + 6k+2= 2(2k² +3k+1), which is even and greater than 2. Since 2 is the only even prime, it follows that n² + 3n+2 is not prime. Which of these proofs are valid (i.e. which of them actually prove the claim)? (a) Both proofs are valid. (b) Proof 1 is valid, but Proof 2 is not. (c) Proof 2 is valid, but Proof 1 is not. (d) Neither proof is valid.

12. Consider the following (true) claim and two possible proofs. Claim. If n is a positive integer, then n² + 3n+2 is not a prime number. Proof 1. Note that n² + 3n+2= (n + 2)(n+1). Since n> 1, we have n + 1>1 and n+2 > 1, so n²+3n+2 has at least two divisors other than 1 and is therefore not prime. Proof 2. If n is even then we can write n = 2k for some positive integer k. Then n² + 3n+2 = 4k² + 6k+2= 2(2k² +3k+1), which is even and greater than 2. Since 2 is the only even prime, it follows that n² + 3n+2 is not prime. Which of these proofs are valid (i.e. which of them actually prove the claim)? (a) Both proofs are valid. (b) Proof 1 is valid, but Proof 2 is not. (c) Proof 2 is valid, but Proof 1 is not. (d) Neither proof is valid.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 49E: Show that if the statement is assumed to be true for , then it can be proved to be true for . Is...

See similar textbooks