명제와 조건 연습문제 — Sundstrom, Mathematical Reasoning · 2.4절 · 2f번

Sundstrom, Mathematical Reasoning · 2.4절 · 2f번영어 원문

For each of the following, use a counterexample to show that the statement is false. Then write the negation of the statement in English, without using symbols for quantifiers.

$\left( {\forall x \in \mathbb{R}} \right)\left( {\tan ^2 x + 1 = \sec ^2 x} \right)$ .

정답·풀이 보기 (영어 원문)

$x = \dfrac{\pi}{2}$ is a counterexample. The negation is: There exists a real number $x$ such that $\tan^2 x + 1 \ne \sec^2 x$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 2.3절영어 원문

Assume the universal set for all variables is $\mathbb{Z}$ , and let $R( {x,\;y,\;z} )$ be the predicate “ $x^2 + y^2 = z^2$ .”

(i) Find two different examples for which $R( {x, y, z})$ is false.

(ii) Find two different examples for which $R( {x, y, z})$ is true.

다 풀었으면 정답 확인

(i) Two examples for which $R( {x, y, z} )$ is false are: $x = 1$ , $y = 1$ , $z = 1$ and $x = 3$ , $y = - 1$ , $z = 5$ .

(ii) Two examples for which $R( {x, y, z} )$ is true are: $x = 3$ , $y = 4$ , $z = 5$ and $x = 5$ , $y = 12$ , $z = 13$ .

이 문제로 이어서 풀기

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