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

Sundstrom, Mathematical Reasoning · 2.4절 · 2b번영어 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.

$\left( {\forall x \in \mathbb{R}} \right)\left( {x^2 > 0} \right)$ .

정답·풀이 보기 (영어 원문)

$x = 0$ is a counterexample. The negation is: There exists a real number $x$ such that $x^2 \leq 0$ .

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

Assume the universal set for all variables is $\mathbb{Z}$ and let $P( x )$ be the predicate “ $x^2 \leq 4$ .”

(i) Find two values of $x$ for which $P( x )$ is false.

(ii) Find two values of $x$ for which $P( x )$ is true.

(iii) Use the roster method to specify the set of all $x$ for which $P( x )$ is true.

다 풀었으면 정답 확인

(i) Two values of $x$ for which $P( x )$ is false are $x = 3$ and $x = -4$ .

(ii) The set of all $x$ for which $P( x )$ is true is the set $\left\{ -2, -1, 0, 1, 2 \right\}$ .

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