명제와 조건 연습문제 — Sundstrom, Mathematical Reasoning · 2.4절 · 10a번

Sundstrom, Mathematical Reasoning · 2.4절 · 10a번영어 원문

In calculus, we define a function $f$ with domain $\mathbb{R}$ to be strictly increasing

provided that for all real numbers $x$ and $y$ , $f\left( x \right) \lt f\left( y \right)$ whenever $x \lt y$ . Complete each of the following sentences using the appropriate symbols for quantifiers:

A function $f$ with domain $\mathbb{R}$ is strictly increasing provided that $\ldots$ .

정답·풀이 보기 (영어 원문)

A function $f$ with domain $\mathbb{R}$ is strictly increasing provided that $\left( \forall x, y \in \mathbb{R} \right) \left[ \left( x \lt y \right) \to \left( f(x) \lt f(y) \right) \right]$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 2.4절영어 원문

Use counterexamples to explain why each of the following statements is false.

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

다 풀었으면 정답 확인

A counterexample is $x = \dfrac{1}{4}$ since $\dfrac{1}{4}$ is positive and $2 \left( \dfrac{1}{4} \right)^2 = \dfrac{1}{8}$ and $\dfrac{1}{8} \leq \dfrac{1}{4}$ .

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