명제와 조건 연습문제 (공통수학2)

1.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

$2 \cdot 7 + 8 = 22$ .

정답·풀이 보기 (영어 원문)

This is a statement.

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

# 명제와 조건계산자동 분류

2.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

$2x + 5y = 7$ .

정답·풀이 보기 (영어 원문)

This is not a statement.

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

# 명제와 조건계산자동 분류

3.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

There are integers $x$ and $y$ such that $2x + 5y = 7$ .

정답·풀이 보기 (영어 원문)

This is a statement.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 2a번

# 명제와 조건계산자동 분류

4.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

Given a line $L$ and a point $P$ not on that line and in the same plane, there is a unique line in that plane through $P$ that does not intersect $L$ .

정답·풀이 보기 (영어 원문)

This is a statement.

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

# 명제와 조건계산자동 분류

5.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

$\left( {a + b} \right)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ .

정답·풀이 보기 (영어 원문)

This is not a statement.

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

# 명제와 조건계산자동 분류

6.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

For all real numbers $a$ and $b$ , $\left( {a + b} \right)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ .

정답·풀이 보기 (영어 원문)

This is a statement.

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

# 명제와 조건계산자동 분류

7.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

Does the equation $3x^2 - 5x - 7 = 0$ have two real number solutions?

정답·풀이 보기 (영어 원문)

This is not a statement.

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

# 명제와 조건계산자동 분류

8.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

If $ABC$ is a right triangle with right angle at vertex $B$ , and if $D$ is the midpoint of the hypotenuse, then the line segment connecting vertex $B$ to $D$ is half the length of the hypotenuse.

정답·풀이 보기 (영어 원문)

This is a statement.

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

# 명제와 조건계산자동 분류

9.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Which of the following sentences are statements? Do not worry about determining the truth value of those that are statements; just determine whether each sentence is a statement or not.

There do not exist three integers $x$ , $y$ , and $z$ such that $x^3 + y^3 = z^3$ .

정답·풀이 보기 (영어 원문)

This is a statement.

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

# 명제와 조건계산자동 분류

10.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

$\left(a+b \right)^2=a^2+b^2$ , for all real numbers $a$ and $b$ .

정답·풀이 보기 (영어 원문)

This proposition is false. A counterexample is $a = 2$ and $b = 1$ . For these values, $\left( {a + b} \right)^2 = 9$ and $a^2 + b^2 = 5$ .

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

# 명제와 조건개념 확인자동 분류

11.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

There are integers $x$ and $y$ such that $2x+5y=41$ .

정답·풀이 보기 (영어 원문)

This proposition is true, as we can see by using $x = 3$ and $y = 7$ . We could also use $x = -2$ and $y = 9$ . There are many other possible choices for $x$ and $y$ .

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 4c번

# 명제와 조건개념 확인자동 분류

12.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

If $x$ is an even integer, then $x^2$ is an even integer.

정답·풀이 보기 (영어 원문)

This proposition appears to be true. Anytime we use an example where $x$ is an even integer, the number $x^2$ is an even integer. However, we cannot claim that this is true based on examples since we cannot list all of the examples where $x$ is an even integer.

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

# 명제와 조건개념 확인자동 분류

13.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

If $x$ and $y$ are odd integers, then $x \cdot y$ is an odd integer.

정답·풀이 보기 (영어 원문)

This proposition appears to be true. Anytime we use an example where $x$ and $y$ are both odd integers, the number $x \cdot y$ is an odd integer. However, we cannot claim that this is true based on examples since we cannot list all of the examples where both $x$ and $y$ are odd integers.

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

# 명제와 조건개념 확인자동 분류

14.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

“If $n$ is a positive integer, then $(n^2-n+41)$ is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for $n=1$ , $n=2$ , $n=3$ , $n=4$ , $n=5$ , and $n=10$ . Then record the results for at least four other values of $n$ . Does this conditional statement appear to be true?

정답·풀이 보기 (영어 원문)

The number $\left( n^2 - n + 41 \right)$ will be a prime number for all examples of $n$ that are less than 41. However, when $n = 41$ , we get

$\begin{aligned}n^2 - n + 41 & = 41^2 - 41 + 41 \\ n^2 - n + 41 & = 41^2\end{aligned}$

So in the case where $n = 41$ , the hypothesis is true (41 is a positive integer) and the conclusion is false $\left( {41^2 \text{ is not prime } } \right)$ . Therefore, 41 is a counterexample that shows the conditional statement is false. There are other counterexamples (such as $n = 42$ , $n = 45$ , and $n = 50$ ), but only one counterexample is needed to prove that the statement is false.

한 번 더 — 비슷한 문제 풀기Sundstrom, Mathematical Reasoning · 2.1절 · 5a번

# 명제와 조건계산자동 분류

15.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Answer each of the following questions.

Is the set of rational numbers closed under addition? Explain.

정답·풀이 보기 (영어 원문)

The set of rational numbers is closed under addition since $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad+bc}{bd}$ .

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

# 명제와 조건계산자동 분류

16.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Answer each of the following questions.

Is the set of integers closed under division? Explain.

정답·풀이 보기 (영어 원문)

The set of integers is not closed under division. For example, $\dfrac{2}{3}$ is not an integer.

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

# 명제와 조건계산자동 분류

17.Sundstrom, Mathematical Reasoning · 1.1절영어 원문

Answer each of the following questions.

Is the set of rational numbers closed under subtraction? Explain.

정답·풀이 보기 (영어 원문)

The set of rational numbers is closed under subtraction since $\dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad-bc}{bd}$ .

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

# 명제와 조건계산자동 분류

18.Sundstrom, Mathematical Reasoning · 1.1절 · 1a번영어 원문

Which of the following sentences are statements?

$3^2 + 4^2 = 5^2$ .

정답·풀이 보기 (영어 원문)

This is a statement.

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

# 명제와 조건계산자동 분류

19.Sundstrom, Mathematical Reasoning · 1.1절 · 1b번영어 원문

Which of the following sentences are statements?

$a^2 + b^2 = c^2$ .

정답·풀이 보기 (영어 원문)

This is not a statement.

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

# 명제와 조건계산자동 분류

20.Sundstrom, Mathematical Reasoning · 1.1절 · 1c번영어 원문

Which of the following sentences are statements?

There exist integers $a$ , $b$ , and $c$ such that $a^2 = b^2 + c^2$ .

정답·풀이 보기 (영어 원문)

This is a statement.

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

# 명제와 조건계산자동 분류