# For all sets A, B and C, show that (A – B) ∩ (C – B) = A – (B ∪ C) - Mathematics

Sum

For all sets A, B and C, show that (A – B) ∩ (C – B) = A – (B ∪ C)

#### Solution

Given: There are three sets A, B and C

To prove: (A – B) ∩ (A – C) = A – (B ∪ C)

Let x ∈ (A – B) ∩ (A – C)

⇒ x ∈ (A – B) and x ∈ (A – C)

⇒ (x ∈ A and x ∉ B) and (x ∈ A and x ∉ C)

⇒ x ∈ A and (x ∉ B and x ∉ C)

⇒ x ∈ A and x ∉ (B ∪ C)

⇒ x ∈ A – (B ∪ C)

⇒ (A – B) ∩ (A – C) ⊂ A – (B ∪ C)  ......(i)

Let y ∈ A – (B ∪ C)

⇒ y ∈ A and y ∉ (B ∪ C)

⇒ y ∈ A and (y ∉ B and y ∉ C)

⇒ (y ∈ A and y ∉ B) and (y ∈ A and y ∉ C)

⇒ y ∈ (A – B) and y ∈ (A – C)

⇒ y ∈ (A – B) ∩ (A – C)

⇒ A – (B ∪ C) ⊂ (A – B) ∩ (A – C) .......(ii)

We know,

P ⊂ Q and Q ⊂ P

⇒ P = Q

From (i) and (ii)

A – (B ∪ C) = (A – B) ∩ (A – C)

Concept: Operations on Sets - Intrdouction of Operations on Sets
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 1 Sets
Exercise | Q 12 | Page 14
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