ITPEC IP April 2022 Question 2
Which of the following is always true for two (2) sets A and B? Here, (X ∩ Y) represents the set that contains only the elements that are in both X and Y (intersection set), and (X ∪ Y) represents the set that contains all of the elements that are in at least one of X or Y (union set).
Explanation
The intersection is a subset of the union — every element of A ∩ B is in both sets, so it is certainly in at least one of them, which puts it in A ∪ B.
An element of A ∩ B belongs to A, and every element of A belongs to A ∪ B. This holds for any two sets, including when the intersection is empty, because the empty set is a subset of every set.
Answer (c)
Why not others:
- •(a) — when A and B share an element, that element is in A ∪ B but not in the complement of A ∩ B
- •(b) — an element of B that is not in A is in A ∪ B but not in A
- •(d) — every element of A ∩ B is in A, so none is in the complement of A; the relation holds only when the intersection is empty
Key rule: For any sets, the intersection is inside each set and each set is inside the union: A ∩ B ⊆ A ⊆ A ∪ B.
Basic Theory · AI-written, may contain errors
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