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Show that the following four conditions are equivalent:
(i) A ⊂ B (ii) A – B = Φ
(iii) A ∪ B = B (iv) A ∩ B = A
Asked by Syed Last Modified
Answer First, we have to show that (i) ⇔ (ii).
Let A ⊂ B
To show: A – B ≠ Φ
If possible, suppose A – B ≠ Φ
This means that there exists x ∈ A, x ≠ B, which is not possible as A ⊂ B.
∴ A – B = Φ
∴ A ⊂ B ⇒ A – B = Φ
Let A – B = Φ
To show: A ⊂ B
Let x ∈ A
Clearly, x ∈ B because if x ∉ B, then A – B ≠ Φ
∴ A – B = Φ ⇒ A ⊂ B
∴ (i) ⇔ (ii)
Let A ⊂ B
To show: 
Clearly, 
Let 
Case I: x ∈ A
∴ 
Case II: x ∈ B
Then,
Conversely, let
Let x ∈ A
∴ A ⊂ B
Hence, (i) ⇔ (iii)
Now, we have to show that (i) ⇔ (iv).
Let A ⊂ B
Clearly
Let x ∈ A
We have to show that
As A ⊂ B, x ∈ B
∴
∴ 
Hence, A = A ∩ B
Conversely, suppose A ∩ B = A
Let x ∈ A
⇒
⇒ x ∈ A and x ∈ B
⇒ x ∈ B
∴ A ⊂ B
Hence, (i) ⇔ (iv).
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