명제와 조건 연습문제 — Sundstrom, Mathematical Reasoning · 1.1절

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 문제Sundstrom, 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?

다 풀었으면 정답 확인

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.

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