Show that the following four conditions are equivalent.(i) A ⊂ B (ii) A – B = Φ(iii) A ∪ B = B (iv) A ∩ B = A
Question
Show that the following four conditions are equivalent.(i) A ⊂ B (ii) A – B = Φ(iii) A ∪ B = B (iv) A ∩ B = A
Solution
(i) A ⊂ B means that every element of A is also an element of B.
(ii) A – B = Φ means that when you subtract all the elements of B from A, you get an empty set. This can only happen if all elements of A are also in B, which is the same as condition (i).
(iii) A ∪ B = B means that the union of A and B is just B. This means that adding all the elements of A to B doesn't change B, which can only happen if all elements of A are already in B. This is the same as conditions (i) and (ii).
(iv) A ∩ B = A means that the intersection of A and B is just A. This means that all elements of A are also in B, which is the same as conditions (i), (ii) and (iii).
Therefore, all four conditions are equivalent.
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