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

Sundstrom, Mathematical Reasoning · 3.3절 · 5a번영어 원문

Prove the following propositions:

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

정답·풀이 보기 (영어 원문)

Use a proof by contradiction. So, we assume that there exist real numbers $x$ and $y$ such that $x$ is rational, $y$ is irrational, and $x + y$ is rational. Since the rational numbers are closed under addition, this implies that $\left( x + y \right) - x$ is a rational number. Since $\left( x + y \right) - x = y$ , 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 · 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.

다 풀었으면 정답 확인

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.

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