명제와 조건 연습문제 (공통수학2)

161.Sundstrom, Mathematical Reasoning · 2.4절영어 원문

An integer $n$ is a multiple of 3 provided that there exists an integer $k$ such that $n = 3k$ .

Without using the symbols for quantifiers, complete the following sentence: “An integer $n$ is not a multiple of 3 provided that ….”

정답·풀이 보기 (영어 원문)

An integer $n$ is not a multiple of 3 provided that for every integer $k$ , $n \ne 3k$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 6e번

# 명제와 조건계산자동 분류

162.Sundstrom, Mathematical Reasoning · 2.4절영어 원문

Write the negation of the statement

$\left( {\forall x \in \mathbb{Z}} \right)\left( {\forall y \in \mathbb{Z}} \right)\left( {x + y = 0} \right)$

in symbolic form and as a sentence written in English.

정답·풀이 보기 (영어 원문)

$\left( {\exists x \in \mathbb{Z}} \right)\left( {\exists y \in \mathbb{Z}} \right)\left( {x + y \ne 0} \right)$ .

There exist integers $x$ and $y$ such that $x + y \ne 0$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 3e번

# 명제와 조건계산자동 분류

163.Sundstrom, Mathematical Reasoning · 2.4절 · 1a번영어 원문

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

$\left(\exists x \in \mathbb{Q} \right) \left( x^2 - 3x - 7 = 0 \right)$ .

정답·풀이 보기 (영어 원문)

There exists a rational number $x$ such that $x^2 - 3x - 7 = 0$ . This statement is false since the solutions of the equation are $x = \dfrac{3 \pm \sqrt{37}}{2}$ , which are irrational numbers.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절

# 명제와 조건계산자동 분류

164.Sundstrom, 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.

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

정답·풀이 보기 (영어 원문)

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.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 6c번

# 명제와 조건계산자동 분류

165.Sundstrom, Mathematical Reasoning · 2.4절 · 1c번영어 원문

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

$\left( \exists m \in \mathbb{N} \right) \left( m^2 \lt 1 \right)$ .

정답·풀이 보기 (영어 원문)

There exists a natural number $m$ such that $m^2 \lt 1$ . This statement is false because if $m$ is a natural number, then $m \geq 1$ and hence, $m^2 \geq 1$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 6b번

# 명제와 조건계산자동 분류

166.Sundstrom, Mathematical Reasoning · 2.4절 · 2a번영어 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.

$\left( {\forall m \in \mathbb{Z}} \right)\left( {m^2 \text{ is even } } \right)$ .

정답·풀이 보기 (영어 원문)

$m = 1$ is a counterexample. The negation is: There exists a natural number $m$ such that $m^2$ is not even or there exists a natural number $m$ such that $m^2$ is odd.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.3절 · 4a번

# 명제와 조건증명·논술자동 분류

167.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 문제 풀기Sundstrom, Mathematical Reasoning · 2.3절

# 명제와 조건증명·논술자동 분류

168.Sundstrom, Mathematical Reasoning · 2.4절 · 2c번영어 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.

For each real number $x$ , $\sqrt{x} \in \mathbb{R}$ .

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 1.1절 · 1k번

# 명제와 조건증명·논술자동 분류

169.Sundstrom, Mathematical Reasoning · 2.4절 · 2d번영어 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.

$\left( \forall m \in \mathbb{Z} \right) \left( \dfrac{m}{3} \in \mathbb{Z} \right)$ .

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 11d번

# 명제와 조건증명·논술자동 분류

170.Sundstrom, Mathematical Reasoning · 2.4절 · 2e번영어 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.

$\left( {\forall a \in \mathbb{Z}} \right)\left( {\sqrt {a^2 } = a} \right)$ .

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.3절 · 4b번

# 명제와 조건증명·논술자동 분류

171.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 문제 풀기Sundstrom, Mathematical Reasoning · 2.3절

# 명제와 조건증명·논술자동 분류

172.Sundstrom, Mathematical Reasoning · 2.4절 · 3a번영어 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.

$\left( {\exists x \in \mathbb{Q}} \right)\left( {x > \sqrt 2 } \right)$ .

정답·풀이 보기 (영어 원문)

There exists a rational number $x$ such that $x > \sqrt{2}$ . The negation is $\left( {\forall x \in \mathbb{Q}} \right) \left( {x \leq \sqrt 2 } \right)$ , which is, For each rational number $x$ , $x \le \sqrt{2}$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 3c번

# 명제와 조건계산자동 분류

173.Sundstrom, Mathematical Reasoning · 2.4절 · 3b번영어 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.

$\left( {\forall x \in \mathbb{Q}} \right)\left( {x^2 - 2 \ne 0} \right)$ .

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 11c번

# 명제와 조건계산자동 분류

174.Sundstrom, Mathematical Reasoning · 2.4절 · 3c번영어 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.

$\left( {\forall x \in \mathbb{Z}} \right)\left( {x\text{ is even or } x\text{ is odd } } \right)$ .

정답·풀이 보기 (영어 원문)

For each integer $x$ , $x$ is even or $x$ is odd. The negation is $\left( \exists x \in \mathbb{Z} \right) \left( x \text{ is odd and $x$ is even } \right)$ , which is, There exists an integer $x$ such that $x$ is odd and $x$ is even.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 3h번

# 명제와 조건계산자동 분류

175.Sundstrom, Mathematical Reasoning · 2.4절 · 3d번영어 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.

$\left( {\exists x \in \mathbb{Q}} \right)\left( {\sqrt 2 \lt x \lt \sqrt 3 } \right)$ .

Note: The sentence “ $\sqrt 2 \lt x \lt \sqrt 3$ ” is actually a conjunction. It means $\sqrt 2 \lt x$ and $x \lt \sqrt 3$ .

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 11a번

# 명제와 조건계산자동 분류

176.Sundstrom, Mathematical Reasoning · 2.4절 · 3e번영어 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.

$\left( {\forall x \in \mathbb{Z}} \right)\left( {\text{ If } x^2 \text{ is odd, then } x\text{ is odd } } \right)$ .

정답·풀이 보기 (영어 원문)

For each integer $x$ , if $x^2$ is odd, then $x$ is odd. The negation is $\left( \exists x \in \mathbb{Z} \right) \left( x^2 \text{ is odd and $x$ is even } \right)$ , which is, There exists an integer $x$ such that $x^2$ is odd and $x$ is even.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 3a번

# 명제와 조건계산자동 분류

177.Sundstrom, Mathematical Reasoning · 2.4절 · 3f번영어 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.

$\left( {\forall n \in \mathbb{N}} \right)$ [If $n$ is a perfect square, then $\left( {2^n - 1} \right)$ is not a prime number].

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.3절

# 명제와 조건계산자동 분류

178.Sundstrom, Mathematical Reasoning · 2.4절 · 3g번영어 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.

$\left( {\forall n \in \mathbb{N}} \right)\left( {n^2 - n + 41} \text{ is a prime number } \right)$ .

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 11b번

# 명제와 조건계산자동 분류

179.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 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 1b번

# 명제와 조건계산자동 분류

180.Sundstrom, Mathematical Reasoning · 2.4절 · 4a번영어 원문

Write each of the following statements as an English sentence that does not use the symbols for quantifiers.

$\left( {\exists m} \in \mathbb{Z} \right)\left( {\exists n} \in \mathbb{Z} \right)\left( {m > n} \right)$

정답·풀이 보기 (영어 원문)

There exist integers $m$ and $n$ such that $m > n$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절

# 명제와 조건계산자동 분류