問62
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の両方に属する部分（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27年度 春期 ITパスポート試験 問62
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 62
Incorrectア
The statement “(A∪B) is a subset of the complement of (A∩B)” is wrong. The union A∪B contains every element that belongs to at least one of A or B, so the elements of the overlapping part A∩B are naturally contained in A∪B as well. That is, A∩B ⊆ A∪B holds, and A∪B never fits inside the complement of A∩B (the part that is not A∩B). If A and B share even one common element, that element is a counterexample.
Incorrectイ
The statement “(A∪B) is a subset of A” is wrong. The union also contains elements that belong only to B, so if B has an element not contained in A, that element is in A∪B but not in A, which breaks the subset condition. The correct inclusion goes the other way: A ⊆ A∪B. It holds only in the special case where B ⊆ A, so it cannot be called a relationship that always holds.
- ウ
正解 The answer is ウ. The intersection A∩B collects only “elements that belong to both A and B”, while the union A∪B collects “elements that belong to at least one of A and B”. Take any element of A∩B: by definition it belongs to both A and B, so it certainly satisfies the condition of belonging to at least one of them, and therefore belongs to A∪B. Hence every element of A∩B is an element of A∪B, that is, A∩B ⊆ A∪B always holds. This is true no matter what sets A and B are, and even when A∩B is the empty set (the empty set is a subset of every set). In a Venn diagram, the overlapping part of circle A and circle B always lies inside the region formed by combining the two circles, which confirms it intuitively. For the same reason A∩B ⊆ A and A∩B ⊆ B also always hold.
Incorrectエ
The statement “(A∩B) is a subset of the complement of A” is wrong. By definition every element of A∩B belongs to A, so it can never fall into the complement of A (the part that is not A). The correct relation is A∩B ⊆ A. It holds only formally when A∩B is the empty set; in the general case where common elements exist it does not hold, so it is not a relationship that always holds.
ここがポイント
- The intersection A∩B consists of the elements belonging to both sets, while the union A∪B consists of the elements belonging to at least one of them. From these definitions, the inclusions A∩B ⊆ A ⊆ A∪B and A∩B ⊆ B ⊆ A∪B always hold.
- The direction of a subset inclusion matters. An intersection is smaller than (or equal to) the original sets, and a union is larger than (or equal to) them. Drawing a Venn diagram and checking the overlapping part against the whole helps you avoid wrong answers.
用語