Question

Show that the following four conditions are equivalent

(i) A ⊂ B

(ii) A – B=φ

(iii) A ∪ B = B

(iv) A ∩ B A.

Answer 1

(i) ⇔ (ii): 

A⊂B ⇔ All elements of A are in B ⇔ A – B = φ 

(ii) ⇔ (iii): 

A-B = φ⇔ All elements of A are in B ⇔ A∪B = B 

(iii) ⇔(iv) 

A∪B = B ⇔ All elements of A are in B ⇔ All elements of A are common in A and B. 

⇔ A∩B = A 

∴ All the four given conditions are equivalent.