명제의 증명과 절대부등식 연습문제 — Sundstrom, Mathematical Reasoning · 3.3절 · 5b번

Sundstrom, Mathematical Reasoning · 3.3절 · 5b번영어 원문

Prove the following propositions:

For all nonzero real numbers $x$ and $y$ , if $x$ is rational and $y$ is irrational, then $xy$ is irrational.

정답·풀이 보기 (영어 원문)

Use a proof by contradiction. So, we assume that there exist nonzero real numbers $x$ and $y$ such that $x$ is rational, $y$ is irrational, and $xy$ is rational. Since the rational numbers are closed under division by nonzero rational numbers, this implies that $\dfrac{xy}{x}$ is a rational number. Since $\dfrac{xy}{x} = y$ , we conclude that $y$ is a rational number and this contradicts the assumption that $y$ is irrational.

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 3.3절 · 2a번영어 원문

Are the following statements true or false? Justify each conclusion.

For all integers $a$ and $b$ , if $a$ is even and $b$ is odd, then $4$ does not divide $\left( a^2 + b^2 \right)$ .

다 풀었으면 정답 확인

This statement is true. Use a proof by contradiction. So assume that there exist integers $a$ and $b$ such that $a$ is even, $b$ is odd, and 4 divides $a^2 + b^2$ . So there exist integers $m$ and $n$ such that

$a = 2m \qquad \text{ and } \qquad a^2 + b^2 = 4n$

. Substitute $a = 2m$ into the second equation and use algebra to rewrite in the form $b^2 = 4(n - m^2)$ . This means that $b^2$ is even and hence, that $b$ is even. This is a contradiction to the assumption that $b$ is odd.

이 문제로 이어서 풀기

# 명제의 증명과 절대부등식증명·논술자동 분류