Question

For any two sets A and B, prove that
(i) B ⊂ A ∪ B

(ii) A ∩ B ⊂ A

(iii) A ⊂ B ⇒ A ∩ B = A

Answer 1

(i) B ⊂ A ∪ B

Now, let us consider an element ‘p’ such that it belongs to B

∴ p ∈ B

 p ∈ B ∪ A

B ⊂ A ∪ B

(ii) A ∩ B ⊂ A

Now, let us consider an element ‘p’ such that it belongs to B

∴ p ∈ A ∩ B

p ∈ A and p ∈ B

A ∩ B ⊂ A

(iii) A ⊂ B ⇒ A ∩ B = A

Then let us consider an element ‘p’ such that it belongs to A ⊂ B.

p ∈ A ⊂ B

Now, x ∈ B

Suppose and p ∈ A ∩ B

x ∈ A and x ∈ B

x ∈ A and x ∈ A (since, A ⊂ B)

∴ (A ∩ B) = A