집합의 뜻과 포함 관계 연습문제 — Sundstrom, Mathematical Reasoning · 2.3절 · 2b번

Sundstrom, Mathematical Reasoning · 2.3절 · 2b번영어 원문

Each of the following sets is defined using the roster method.

$\begin{aligned}A & = \left\{1, 4, 9, 16, 25, \ldots \right\} & B & = \left\{\ldots, -\pi^4, -\pi^3, -\pi^2, -\pi, -1 \right\} \\ C & = \left\{3, 9, 15, 21, 27, \ldots \right\} & D & = \left\{0, 4, 8, \ldots, 96, 100 \right\}\end{aligned}$

Use set builder notation to describe each set.

정답·풀이 보기 (영어 원문)

Possible answers:

$\begin{aligned}A & = \left\{ n^2 \mid n \in \mathbb{N} \right\} \\ B & = \left\{ -\pi^n \mid n \text{ ~is a nonnegative integer } \right\} \\ C & = \left\{ 6n + 3 \mid n \text{ ~is a nonnegative integer } \right\} = \left\{6n - 3 \mid n \in \mathbb{N} \right\} \\ D & = \left\{ 4n \mid n \in \mathbb{Z} \text{ ~and~ } 0 \leq n \leq 25 \right\}\end{aligned}$

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 5.2절 · 15b번영어 원문

Are the following biconditional statements true or false? Justify your conclusion. If a biconditional statement is found to be false, you should clearly determine if one of the conditional statements within it is true and provide a proof of this conditional statement.

For all subsets $A$ and $B$ of some universal set $U$ , $A \subseteq B$ if and only if $A \cup B = B$ .

다 풀었으면 정답 확인

To prove “If $A \subseteq B$ , then $A \cup B = B$ ,” first note that if $x \in B$ , then $x \in A \cup B$ and, hence, $B \subseteq A \cup B$ . Now let $x \in A \cup B$ and note that since $A \subseteq B$ , if $x \in A$ , then $x \in B$ . Use this to argue that under the assumption that $A \subseteq B$ , $A \cup B \subseteq B$ . To prove “If $A \cup B = B$ , then $A \subseteq B$ ,” start with, Let $x \in A$ and use this assumption to prove that $x$ must be an element of $B$ .

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# 집합의 뜻과 포함 관계계산자동 분류