Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B
(Hints A = A ∩ ( A ∪ X ) , B = B ∩ ( B ∪ X ) and use Distributive law )

Explanation:

Let A and B be two sets such that A ∩ X = B ∩ X = Φ and A υ X = B υ X for some set X.

We know that

A = A ∩ (A υ X)

= A ∩ (B υ X) [∵ A υ X = B υ X]

= (A ∩ B) υ (A ∩ X ) [∵ (A ∩ X) = Φ]....(1)

= (A ∩ B) υ Φ

= (A ∩ B)

Now,

B = B ∩ (B υ X)

= B ∩ (A υ X) [∵ A υ X = B υ X]

= (B ∩ A) υ ( B ∩ X)

= (B ∩ A) υ Φ [∵ ( B ∩ X ) = Φ]....(2)

= (A ∩ B)

From (1) and (2), we obtain A = B.

Hence proved

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NCERT Solutions Class 11 Maths Chapter 1 Exercise ME Question 11

Let A and B be sets. If A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X, show that A = B. (Hints A = A ∩ ( A ∪ X ) , B = B ∩ ( B ∪ X ) and use Distributive law )

Summary:

Given that A ∩ X = B ∩ X = φ and A ∪ X = B ∪ X for some set X . We have proved that A = B