집합과 명제 전체 연습문제 (공통수학2)

141.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.

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

정답·풀이 보기 (영어 원문)

For each real number $x$ , $\tan ^2 x + 1 = \sec ^2 x$ . - $\left( \exists x \in \mathbb{R}\right) \left( \tan ^2 x + 1 \ne \sec ^2 x \right)$ . - There exists a real number $x$ such that $\tan ^2 x + 1 \ne \sec ^2 x$ .

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

# 명제와 조건계산자동 분류

142.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.

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

정답·풀이 보기 (영어 원문)

There exists a rational number $x$ such that $x^2 - 3x - 7 = 0$ . - $(\forall x \in \mathbb{Q}) \left( x^2 - 3x - 7 \ne 0 \right)$ . - For each rational number $x$ , $x^2 - 3x - 7 \ne 0$ .

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

# 명제와 조건계산자동 분류

143.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.

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

정답·풀이 보기 (영어 원문)

There exists a real number $x$ such that $x^2 + 1 = 0$ . - $(\forall x \in \mathbb{R}) \left( \text{ } x^2 + 1 \ne 0 \right)$ . - For each real number $x$ , $x^2 + 1 \ne 0$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 2b번

# 명제와 조건계산자동 분류

144.Sundstrom, Mathematical Reasoning · 2.4절영어 원문

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

For each integer $n$ , $\left( n^2 + n + 1 \right)$ is a prime number.

정답·풀이 보기 (영어 원문)

A counterexample is $n = 4$ since $4^2 + 4 + 1 = 21$ , and 21 is not prime.

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

# 명제와 조건계산자동 분류

145.Sundstrom, 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 $2x^2 > x$ .

정답·풀이 보기 (영어 원문)

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 $\dfrac{1}{8} \leq \dfrac{1}{4}$ .

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

# 명제와 조건계산자동 분류

146.Sundstrom, Mathematical Reasoning · 2.4절영어 원문

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

Write this definition in symbolic form using quantifiers by completing the following:

An integer

n

is a multiple of 3 provided that

.

정답·풀이 보기 (영어 원문)

An integer $n$ is a multiple of 3 provided that $\left( \exists k \in \mathbb{Z} \right) \left( n = 3k \right)$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 7번

# 명제와 조건계산자동 분류

147.Sundstrom, Mathematical Reasoning · 2.4절영어 원문

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

Give several examples of integers (including negative integers) that are multiples of 3.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 4e번

# 명제와 조건계산자동 분류

148.Sundstrom, Mathematical Reasoning · 2.4절영어 원문

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

Give several examples of integers (including negative integers) that are not multiples of 3.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

149.Sundstrom, Mathematical Reasoning · 2.4절영어 원문

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

Use the symbolic form of the definition of a multiple of 3 to 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 $\left( \forall k \in \mathbb{Z} \right) \left( n \ne 3k \right)$ .

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

# 명제와 조건계산자동 분류

150.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

151.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

152.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

153.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

154.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

155.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

156.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

157.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

158.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

159.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

160.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

집합과 명제 전체 문제