명제의 증명과 절대부등식 연습문제 — Sundstrom, Mathematical Reasoning · 3.3절 · 6a번

Sundstrom, Mathematical Reasoning · 3.3절 · 6a번영어 원문

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

For each positive real number $x$ , if $x$ is irrational, then $x^2$ is irrational.

정답·풀이 보기 (영어 원문)

This statement is false. A counterexample is $x = \sqrt{2}$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 3.3절 · 6b번영어 원문

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

For each positive real number $x$ , if $x$ is irrational, then $\sqrt x$ is irrational.

다 풀었으면 정답 확인

This statement is true since the contrapositive is true. The contrapositive is:

For any real number

, if

\sqrt{x}

is rational, then

is rational.

If there exist integers $a$ and $b$ with $b \ne 0$ such that $\sqrt{x} = \dfrac{a}{b}$ , then $x^2 = \dfrac{a^2}{b^2}$ and hence, $x^2$ is rational.