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

Sundstrom, Mathematical Reasoning · 3.1절영어 원문

Use a counterexample to prove the following statement is false.

For all integers

and

, if 5 divides

or 5 divides

, then 5 divides

(5a + b)

정답·풀이 보기 (영어 원문)

A counterexample for this statement will be values of $a$ and $b$ for which 5 divides $a$ or 5 divides $b$ , and 5 does not divide $5a + b$ . One counterexample for the statement is $a = 5$ and $b = 1$ . For these values, the hyothesis is true since 5 divides $a$ and the conclusion is false since $5a + b = 26$ and 5 does not divide 26.

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 3.3절 · 1a번영어 원문

This exercise is intended to provide another rationale as to why a proof by contradiction works. Suppose that we are trying to prove that a statement $P$ is true. Instead of proving this statement, assume that we prove that the conditional statement “If $\neg P$ , then $C$ ” is true, where $C$ is some contradiction. Recall that a contradiction is a statement that is always false.

In symbols, write a statement that is a disjunction and that is logically equivalent to $\neg P \to C$ .

다 풀었으면 정답 확인

$P \vee C$

이 문제로 이어서 풀기

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