집합의 연산 연습문제 — Sundstrom, Mathematical Reasoning · 5.3절

Sundstrom, Mathematical Reasoning · 5.3절영어 원문

Draw two general Venn diagrams for the sets $A$ , $B$ , and $C$ . On one, shade the region that represents $\left( {A \cup B} \right) - C$ , and on the other, shade the region that represents $\left( {A - C} \right) \cup \left( {B - C} \right)$ . Explain carefully how you determined these regions and why they indicate that $\left( {A \cup B} \right) - C = \left( {A - C} \right) \cup \left( {B - C} \right)$ .

정답·풀이 보기 (영어 원문)

Using our standard configuration for a Venn diagram with three sets, regions 1, 2, and 3 are the regions that are shaded for both $\left( {A \cup B} \right) - C$ and $\left( {A - C} \right) \cup \left( {B - C} \right)$ .

한 번 더 — 비슷한 문제Sundstrom, Mathematical Reasoning · 5.3절 · 6a번영어 원문

Let $A$ , $B$ , and $C$ be subsets of some universal set $U$ . Prove or disprove each of the following:

$\left( A \cap B \right) - C = \left( A - C \right) \cap \left( B - C \right)$

다 풀었으면 정답 확인

Using the algebra of sets, we see that

$\begin{aligned}(A - C) \cap (B - C) & = (A \cap C^c) \cap (B \cap C^c) \\ & = (A \cap B) \cap (C^c \cap C^c) \\ & = (A \cap B) \cap C^c \\ & = (A \cap B) - C\end{aligned}$

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