명제와 조건 연습문제 (공통수학2)

101.Sundstrom, Mathematical Reasoning · 2.3절 · 4e번영어 원문

Use the roster method to specify the truth set for each of the following open sentences. The universal set for each open sentence is the set of integers $\mathbb{Z}$ .

$n$ is an even integer that is greater than 10.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

102.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 문제 풀기Sundstrom, Mathematical Reasoning · 2.4절 · 2f번

# 명제와 조건계산자동 분류

103.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.

$\left( \forall x \in \mathbb{R} \right) \left[ \sin ( {2x} ) = 2 ( {\sin x} ) ( {\cos x} ) \right]$ .

정답·풀이 보기 (영어 원문)

For each real number $x$ , $\sin ( {2x} ) = 2( {\sin x} )( {\cos x} )$ . - $(\exists x \in \mathbb{R})( \sin ( {2x} ) \ne 2( {\sin x} )( {\cos x} ))$ . - There exists a real number $x$ such that $\sin \left( {2x} \right) \ne 2\left( {\sin x} \right)\left( {\cos x} \right)$ .

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

# 명제와 조건계산자동 분류

104.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

105.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

106.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

107.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

108.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

109.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

110.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

111.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

112.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

113.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

114.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

115.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

116.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

117.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

118.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

119.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

120.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