집합과 명제 전체 연습문제 (공통수학2)

121.Sundstrom, Mathematical Reasoning · 2.1절 · 12a번영어 원문

For statements $P$ , $Q$ , and $R$ :

Show that $\left[ (P \to Q) \wedge P \right] \to Q$ is a tautology.

Note: In symbolic logic, this is an important logical argument form called modus ponens.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건증명·논술자동 분류

122.Sundstrom, Mathematical Reasoning · 2.1절 · 12b번영어 원문

For statements $P$ , $Q$ , and $R$ :

Show that $\left[ (P \to Q) \wedge (Q \to R) \right] \to (P \to R)$ is a tautology.

Note: In symbolic logic, this is an important logical argument form called syllogism.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건증명·논술자동 분류

123.Sundstrom, Mathematical Reasoning · 2.2절 · 1a번영어 원문

Write the converse and contrapositive of each of the following conditional statements.

If $a = 5$ , then $a^2 = 25$ .

정답·풀이 보기 (영어 원문)

Converse: If $a^2 = 25$ , then $a = 5$ . Contrapositive: If $a^2 \ne 25$ , then $a \ne 5$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.2절 · 1b번

# 명제와 조건계산자동 분류

124.Sundstrom, Mathematical Reasoning · 2.2절 · 1b번영어 원문

Write the converse and contrapositive of each of the following conditional statements.

If it is not raining, then Laura is playing golf.

정답·풀이 보기 (영어 원문)

Converse: If Laura is playing golf, then it is not raining. Contrapositive: If Laura is not playing golf, then it is raining.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.2절 · 1c번

# 명제와 조건계산자동 분류

125.Sundstrom, Mathematical Reasoning · 2.2절 · 1c번영어 원문

Write the converse and contrapositive of each of the following conditional statements.

If $a \ne b$ , then $a^4 \ne b^4$ .

정답·풀이 보기 (영어 원문)

Converse: If $a^4 \ne b^4$ , then $a \ne b$ . Contrapositive: If $a^4 = b^4$ , then $a = b$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.2절 · 3b번

# 명제와 조건계산자동 분류

126.Sundstrom, Mathematical Reasoning · 2.2절 · 1d번영어 원문

Write the converse and contrapositive of each of the following conditional statements.

If $a$ is an odd integer, then $3a$ is an odd integer.

정답·풀이 보기 (영어 원문)

Converse: If $3a$ is an odd integer, then $a$ is an odd integer. Contrapositive: If $3a$ is an even integer, then $a$ is an even integer.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.2절 · 1a번

# 명제와 조건계산자동 분류

127.Sundstrom, Mathematical Reasoning · 2.2절 · 3a번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

We will win the first game and we will win the second game.

정답·풀이 보기 (영어 원문)

We will not win the first game or we will not win the second game.

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

# 명제와 조건활용자동 분류

128.Sundstrom, Mathematical Reasoning · 2.2절 · 3b번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

They will lose the first game or they will lose the second game.

정답·풀이 보기 (영어 원문)

They will not lose the first game and they will not lose the second game.

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

# 명제와 조건활용자동 분류

129.Sundstrom, Mathematical Reasoning · 2.2절 · 3c번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If you mow the lawn, then I will pay you $20.

정답·풀이 보기 (영어 원문)

You mow the lawn and I will not pay you $20.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.2절 · 3a번

# 명제와 조건계산자동 분류

130.Sundstrom, Mathematical Reasoning · 2.2절 · 3d번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If we do not win the first game, then we will not play a second game.

정답·풀이 보기 (영어 원문)

We do not win the first game and we will play a second game.

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

# 명제와 조건활용자동 분류

131.Sundstrom, Mathematical Reasoning · 2.2절 · 3e번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

I will wash the car or I will mow the lawn.

정답·풀이 보기 (영어 원문)

I will not wash the car and I will not mow the lawn.

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

# 명제와 조건계산자동 분류

132.Sundstrom, Mathematical Reasoning · 2.2절 · 3f번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If you graduate from college, then you will get a job or you will go to graduate school.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

133.Sundstrom, Mathematical Reasoning · 2.2절 · 3g번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If I play tennis, then I will wash the car or I will do the dishes.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.2절 · 1a번

# 명제와 조건계산자동 분류

134.Sundstrom, Mathematical Reasoning · 2.2절 · 3h번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

If you clean your room or do the dishes, then you can go to see a movie.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

135.Sundstrom, Mathematical Reasoning · 2.2절 · 3i번영어 원문

Write a useful negation of each of the following statements. Do not leave a negation as a prefix of a statement. For example, we would write the negation of “I will play golf and I will mow the lawn” as “I will not play golf or I will not mow the lawn.”

It is warm outside and if it does not rain, then I will play golf.

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

136.Sundstrom, Mathematical Reasoning · 2.2절 · 4a번영어 원문

Use truth tables to establish each of the following logical equivalencies dealing with biconditional statements:

$\left( {P \leftrightarrow Q} \right) \equiv \left( {P \to Q} \right) \wedge \left( {Q \to P} \right)$

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

137.Sundstrom, Mathematical Reasoning · 2.2절 · 4b번영어 원문

Use truth tables to establish each of the following logical equivalencies dealing with biconditional statements:

$\left( {P \leftrightarrow Q} \right) \equiv \left( {Q \leftrightarrow P} \right)$

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

138.Sundstrom, Mathematical Reasoning · 2.2절 · 4c번영어 원문

Use truth tables to establish each of the following logical equivalencies dealing with biconditional statements:

$\left( {P \leftrightarrow Q} \right) \equiv \left( {\neg P \leftrightarrow \neg Q} \right)$

원본 교재에 이 문제의 정답은 실려 있지 않습니다.

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

# 명제와 조건계산자동 분류

139.Sundstrom, Mathematical Reasoning · 2.2절 · 7a번영어 원문

Use previously proven logical equivalencies to prove each of the following logical equivalencies about conditionals with conjunctions:

$\left[ {\left( {P \wedge Q} \right) \to R} \right] \equiv \left( {P \to R} \right) \vee \left( {Q \to R} \right)$

정답·풀이 보기 (영어 원문)

In this case, it may be better to work with the right side first.

$\begin{aligned}\left( {P \to R} \right) \vee \left( {Q \to R} \right) & \equiv \left( \neg P \vee R \right) \vee \left( \neg Q \vee R \right) \\ & \equiv \left( \neg P \vee \neg Q \right) \vee \left( R \vee R \right) \\ & \equiv \left( \neg P \vee \neg Q \right) \vee R \\ & \equiv \neg \left( P \wedge Q \right) \vee R \\ & \equiv \left( P \wedge Q \right) \to R\end{aligned}$

.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 1.1절 · 6b번

# 명제와 조건증명·논술자동 분류

140.Sundstrom, Mathematical Reasoning · 2.2절 · 7b번영어 원문

Use previously proven logical equivalencies to prove each of the following logical equivalencies about conditionals with conjunctions:

$\left[ {P \to \left( {Q \wedge R} \right)} \right] \equiv \left( {P \to Q} \right) \wedge \left( {P \to R} \right)$

정답·풀이 보기 (영어 원문)

In this case, we start with the left side.

$\begin{aligned}\left[ P \to \left( Q \wedge R \right) \right] & \equiv \neg P \vee \left( Q \wedge R \right) \\ & \equiv \left( \neg P \vee Q \right) \wedge \left( \neg P \vee R \right) \\ & \equiv \left( P \to Q \right) \wedge \left( P \to R \right)\end{aligned}$

.

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

# 명제와 조건증명·논술자동 분류

집합과 명제 전체 문제