명제의 증명과 절대부등식 연습문제 — Sundstrom, Mathematical Reasoning · 3.3절 · 3a번

Sundstrom, Mathematical Reasoning · 3.3절 · 3a번영어 원문

Consider the following statement:

For each positive real number

r

, if

r^2 = 18

, then

r

is irrational.

If you were setting up a proof by contradiction for this statement, what would you assume? Carefully write down all conditions that you would assume.

정답·풀이 보기 (영어 원문)

We would assume that there exists a positive real number $r$ such that $r^2 = 18$ and $r$ is a rational number.

한 번 더 — 비슷한 문제Sundstrom, 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.

다 풀었으면 정답 확인

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.

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