問69
頻出度★★☆
二つの集合 A と B について，常に成立する関係を記述したものはどれか。ここで，(X∩Y) は，X と Y の共通部分（積集合），(X∪Y) は，X 又は Y の少なくとも一方に属する部分（和集合）を表す。
平成22年度 春期 ITパスポート試験 問69
テクノロジ系基礎理論 › 離散数学
This question's explanation and your answers so far.
解説
問 69
Incorrectア
The set of things that are not A — the complement of A — contains only elements that do not belong to A. The elements of (A∩B), on the other hand, all belong to A, so the two share no elements at all. For example, if A={1,2} and B={2,3}, then (A∩B)={2}; since 2 is an element of A, it is not contained in the complement of A, so the subset relationship does not hold. It happens to hold only when (A∩B) is the empty set, so it cannot be said to always hold.
- イ
正解 The correct answer is イ, “(A∩B) is a subset of A.” By the definition given in the question, the intersection (A∩B) is the common part of A and B — that is, the set formed by collecting only the elements that belong to both A and B at the same time. Every one of those elements is, without exception, also an element of A, so all elements of (A∩B) are contained in A. This is exactly the definition of a subset, so (A∩B) ⊆ A always holds regardless of what A and B contain. In a Venn diagram, this corresponds to the fact that the overlapping part of circle A and circle B is always drawn inside circle A. Note that when A and B have no common elements, (A∩B) becomes the empty set, and since the empty set is a subset of any set, the relationship holds in that case as well. For the same reason, (A∩B) ⊆ B also always holds.
Incorrectウ
The inclusion is the other way around. The union (A∪B) gathers every element that belongs to at least one of A or B, so it is the widest range, while the intersection (A∩B) contains only the elements belonging to both, so it is the narrowest range; therefore the relationship that always holds is (A∩B) ⊆ (A∪B). If A={1,2} and B={2,3}, then (A∪B)={1,2,3} and (A∩B)={2}, and since 1 and 3 are not in (A∩B), it does not hold. It holds only in the special case where A and B are exactly the same.
Incorrectエ
(A∪B) also contains elements that belong only to B, so if such an element lies outside A, the union is not a subset of A. If A={1,2} and B={2,3}, then (A∪B)={1,2,3}, and 3 does not belong to A, so it does not hold. This holds only when B is a subset of A; it is not a relationship that always holds for arbitrary A and B.
ここがポイント
- The intersection (A∩B) is the collection of elements that belong to both A and B, the narrowest range, while the union (A∪B) is the collection of elements that belong to at least one of A or B, the widest range. Two chains of inclusion always hold: (A∩B) ⊆ A ⊆ (A∪B), and (A∩B) ⊆ B ⊆ (A∪B).
- When you are asked whether a relationship always holds, the sure approach is to draw a Venn diagram, or to plug in a concrete example with only partial overlap, such as A={1,2} and B={2,3}, and look for a counterexample. An option that holds only under a specific condition (A and B being identical, one being contained in the other, or the common part being empty) should be judged wrong.
用語