Question

Let A and B be two sets. Using properties of sets prove that:

(i) ∩ ′ = ∅ ⇒ ⊂ (ii) ′ ∪ =∪⇒ ⊂ ​

Answer 1

SOLUTION

TO PROVE

Let A and B be two sets. Using properties of sets prove that:

(i) A ∩ B′ = Φ ⇒ A ⊂ B

(ii) A′ ∪ B = U ⇒ A ⊂ B

PROOF

Here it is given that A and B be two sets

Φ : Empty set , the set containing no element

U : Universal Set

A' = { x : x ∉ A }

(i) Here it is given that A ∩ B′ = Φ

We know that

B ∪ B ' = U

⇒ A ∩ ( B ∪ B' ) = A ∩ U

⇒ ( A ∩ B ) ∪ ( A ∩ B′ ) = A

⇒ ( A ∩ B ) ∪ Φ = A

⇒ ( A ∩ B ) = A

⇒ A ⊂ B

(ii) Here it is given that A′ ∪ B = U

A′ ∪ B = U

⇒ A ∩ ( A' ∪ B ) = A ∩ U

⇒ ( A ∩ A' ) ∪ ( A ∩ B ) = A

⇒ Φ ∪ ( A ∩ B ) = A

⇒ ( A ∩ B ) = A

⇒ A ⊂ B

Hence proved

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