Show that the following four conditions are equivalent: (i) A ⊂ B (ii) A - B = Φ (iii) A ∪ B = B (iv) A ∩ B = A

Show that the following four conditions are equivalent:
(i) A ⊂ B (ii) A - B = Φ (iii) A ∪ B = B (iv) A ∩ B = A 

Answer
First, we shall try to prove A ⊂ B ⇔ A - B = Φ
Given A ⊂ B
To prove A - B = Φ
A ⊂ B so that A ∩ B = A
LHS
=A - B
= A – (A ∩ B)
= A-A
= Φ
RHS
Given A - B = Φ
To prove A ⊂ B
Let x ∈ A
Given that A - B = Φ so all element of A must be in set B
Therefore, x ∈ B
So that A ⊂ B
Hence proved
Similarly you can solve all other parts.