명제의 증명과 절대부등식 연습문제 — Sundstrom, Mathematical Reasoning · 3.1절 · 2a번

Sundstrom, Mathematical Reasoning · 3.1절 · 2a번영어 원문

For each of the following, use a counterexample to prove the statement is false.

For each odd natural number $n$ , if $n > 3$ , then 3 divides $\left( n^2 - 1 \right)$ .

정답·풀이 보기 (영어 원문)

The natural number $n = 9$ is a counterexample since $n$ is odd, $n > 3$ , $n^2 - 1 = 80$ and 3 does not divide 80.

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 3.1절 · 3d번영어 원문

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows it is false.

For each integer $n$ , if 7 divides $\left( n^2 - 4 \right)$ , then 7 divides $(n - 2)$ .

다 풀었으면 정답 확인

This statement is false. One counterexample is $n = 5$ . Since $n^2 - 4 = 21$ and $n - 2 = 3$ , this is an example where the hypothesis of the conditional statement is true and the conclusion is false.

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