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Question
Show that the following four conditions are equivalent:
- A ⊂ B
- A – B = Φ
- A ∪ B = B
- A ∩ B = A
Solution
(i) ⇒ (ii)
A - B = {x : x ∈ A and x ∉ B}
Since A ⊂ B
∴ A - B = Φ
(ii) ⇒ (iii)
A - B = Φ ⇒ A ⊂ B ⇒ A ∪ B = B
(iii) ⇒ (iv)
A ∪ B = B ⇒ A ⊂ B ⇒ A ∩ B = A
(iv) ⇒ (i)
A ∩ B = A ⇒ A ⊂ B
Thus (i) ⇔ (ii) ⇔ (iii) ⇔ (iv).
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