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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) - CBSE (Arts) Class 11 - Mathematics

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Question

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)

Solution

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

To show: A = B

It can be seen that

A = A ∩ (A ∪ X) = A ∩ (B ∪ X) [A ∪ X = B ∪ X]

= (A ∩ B) ∪ (A ∩ X) [Distributive law]

= (A ∩ B) ∪ Φ [A ∩ X = Φ]

= A ∩ B … (1)

Now, B = B ∩ (B ∪ X)

= B ∩ (A ∪ X) [A ∪ X = B ∪ X]

= (B ∩ A) ∪ (B ∩ X) [Distributive law]

= (B ∩ A) ∪ Φ [B ∩ X = Φ]

= B ∩ A

= A ∩ B … (2)

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

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Solution 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) Concept: Operations on Sets - Intersection of Sets.
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