명제의 증명과 절대부등식 연습문제 — Sundstrom, Mathematical Reasoning · 3.1절 · 3d번

Sundstrom, 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 $(n - 2)$ .

정답·풀이 보기 (영어 원문)

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.

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 3.1절 · 3g번영어 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 all integers $a$ , $b$ , and $d$ with $d \ne 0$ , if $d$ divides both $a-b$ and $a+b$ , then $d$ divides $a$ .

다 풀었으면 정답 확인

This statement is false. One counterexample is $a = 7$ , $b = 1$ , and $d = 2$ . Why is this a counterexample?

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