Q. 64.2( 12 Votes )

# If A and B are subsets of the universal set U, then show that

(i) A ⊂ A ∪ B

(ii) A ⊂ B ⇔ A ∪ B = B

(iii) (A ∩ B) ⊂ A

Answer :

(i) **Given:** A and B are two subsets

**To prove:** A ⊂ A ∪ B

Let x ∈ A

⇒ x ∈ A or x ∈ B

⇒ x ∈ A ∪ B

⇒ A ⊂ A ∪ B

**Hence Proved**

(ii) **Given:** A and B are two sets

**To prove:** A ⊂ B ⇔ A ∪ B = B

Let x ∈ A ∪ B

⇒ x ∈ A or x ∈ B

⇒ x ∈ B {∵ A ⊂ B}

⇒ A ∪ B ⊂ B…………(1)

We know,

B ⊂ A ∪ B {this is always true}…………(2)

From (1) and (2):

A ∪ B = B

Now,

Let y ∈ A

⇒ y ∈ A ∪ B

⇒ y ∈ B {∵ A ∪ B = B}

⇒ A ⊂ B

So,

A ⊂ B ⇔ A ∪ B = B

**Hence Proved**

(iii) **Given:** A and B are two subsets

**To prove:** (A ∩ B) ⊂ A

Let x ∈ A ∩ B

⇒ x ∈ A and x ∈ B

⇒ x ∈ A

⇒ A ∩ B ⊂ A

**Hence Proved**

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