집합의 뜻과 포함 관계 연습문제 — Sundstrom, Mathematical Reasoning · 5.2절 · 15b번

Sundstrom, 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 $A \cup B = B$ .

정답·풀이 보기 (영어 원문)

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 $B$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 5.2절 · 12a번영어 원문

Let $A$ , $B$ , and $C$ be subsets of a universal set $U$ . Prove:

If $A \subseteq B$ , then $A \cap C \subseteq B \cap C$ .

다 풀었으면 정답 확인

Let $x \in A \cap C$ . Then $x \in A$ and $x \in C$ . Since we are assuming that $A \subseteq B$ , we see that $x \in B$ and $x \in C$ . This proves that $A \cap C \subseteq B \cap C$ .

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# 집합의 뜻과 포함 관계개념 확인자동 분류