명제와 조건 연습문제 — Sundstrom, Mathematical Reasoning · 2.4절

Sundstrom, Mathematical Reasoning · 2.4절영어 원문

For each of the following statements

(1) Write the statement in the form of an English sentence that does not use the symbols for quantifiers. - (2) Write the negation of the statement in a symbolic form that does not use the negation symbol. - (3) Write the negation of the statement in the form of an English sentence that does not use the symbols for quantifiers.

$\left( \forall a \in \mathbb{R}\right) \left( a + 0 = a \right)$ .

정답·풀이 보기 (영어 원문)

For each real number $a$ , $a + 0 = a$ . - $(\exists a \in \mathbb{R})( a + 0 \ne a)$ . - There exists a real number $a$ such that $a + 0 \ne a$ .

한 번 더 — 비슷한 문제Sundstrom, 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.

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

다 풀었으면 정답 확인

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

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