(i) A ⊂ B
To prove that the following four statements are equivalent, we need to prove (i)=(ii), (ii)=(iii), (iii)=(iv), (iv)=(v)
Now, firstly let us prove (i)=(ii)
As we know, A–B = {x ∈ A: x ∉ B} as A ⊂ B,
Therefore, each element of A is an element of B,
∴ A–B = ϕ
Thus, (i)=(ii)
(ii) A – B = ϕ
As we need to show that (ii)=(iii)
On assuming A–B = ϕ
To prove: A∪B = B
∴ Every element of A is an element of B
Therefore, A ⊂ B and so A∪B = B
Thus, (ii)=(iii)
(iii) A ∪ B = B
As we need to show that (iii)=(iv)
On assuming A ∪ B = B
To prove: A ∩ B = A.
∴ A⊂ B and so A ∩ B = A
Thus, (iii)=(iv)
(iv) A ∩ B = A
Finally, we need to show (iv)=(i)
On assuming A ∩ B = A
To prove: A ⊂ B
Thus, A ∩ B = A, therefore A ⊂ B
Thus, (iv) = (i)
