명제와 조건 연습문제 — Sundstrom, Mathematical Reasoning · 2.4절 · 3h번

Sundstrom, Mathematical Reasoning · 2.4절 · 3h번영어 원문

For each of the following statements

Write the statement as an English sentence that does not use the symbols for quantifiers. - Write the negation of the statement in symbolic form in which the negation symbol is not used. - Write a useful negation of the statement in an English sentence that does not use the symbols for quantifiers.

$\left( {\exists x \in \mathbb{R}} \right) \left( {\cos ( {2x} ) = 2( {\cos x} )} \right)$ .

정답·풀이 보기 (영어 원문)

There exists a real number $x$ such that $\cos \left( 2x \right) = 2 \left( \cos x \right)$ . The negation is $\left( {\forall x \in \mathbb{R}} \right) \left( \cos \left( 2x \right) \ne 2 \left( \cos x \right) \right)$ , which is, For each real number $x$ , $\cos \left( 2x \right) \ne 2 \left( \cos x \right)$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 2.4절 · 1b번영어 원문

For each of the following, write the statement as an English sentence and then explain why the statement is false.

$\left( \exists x \in \mathbb{R} \right) \left( x^2 + 1 = 0 \right)$ .

다 풀었으면 정답 확인

There exists a real number $x$ such that $x^2 + 1 = 0$ . This statement is false since the only solutions of the equation are $i$ and $-i$ , which are not real numbers.

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