If A and B are subsets of the universal set U,

Solution:

(i) According to the question,

A and B are two subsets

To prove: A ⊂ A ∪ B

Proof:

Let x ∈ A

⇒ x ∈ A or x ∈ B

⇒ x ∈ A ∪ B

⇒ A ⊂ A ∪ B

Hence Proved

(ii) According to the question,

A and B are two subsets

To prove: A ⊂ B ⇔ A ∪ B = B

Proof:

Let x ∈ A ∪ B

⇒ x ∈ A or x ∈ B

Since, A ⊂ B, we get,

⇒ x ∈ B

⇒ A ∪ B ⊂ B …(i)

We know that,

B ⊂ A ∪ B …(ii)

From equations (i) and (ii),

We get,

A ∪ B = B

Now,

Let y ∈ A

⇒ y ∈ A ∪ B

Since, A ∪ B = B, we get,

⇒ y ∈ B }

⇒ A ⊂ B

So,

A ⊂ B ⇔ A ∪ B = B

Hence Proved

(iii) According to the question,

A and B are two subsets

To prove: (A ∩ B) ⊂ A

Proof:

Let x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

⇒ x ∈ A

⇒ A ∩ B ⊂ A

Hence Proved