NCERT Class XI Mathematics - Sets - Solutions
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Question : 65
Total: 73
Show that for any sets A and B,
A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)
A = (A ∩ B) ∪ (A – B) and A ∪ (B – A) = (A ∪ B)
Solution:
(A ∩ B) ∪ (A – B) = (A ∩ B) ∪ (A ∩ B′)
= A ∩ (B ∪ B′) (By distributive law)
= A ∩ U = A
Hence, A = (A ∩ B) ∪ (A – B)
Also A ∪ (B – A) = A ∪ (B ∩ A′)
= (A ∪ B) ∩ (A ∪ A′) (By distributive law)
= (A ∪ B) ∩ U = A ∪ B
Hence, A ∪ (B – A) = A ∪ B.
= A ∩ (B ∪ B′) (By distributive law)
= A ∩ U = A
Hence, A = (A ∩ B) ∪ (A – B)
Also A ∪ (B – A) = A ∪ (B ∩ A′)
= (A ∪ B) ∩ (A ∪ A′) (By distributive law)
= (A ∪ B) ∩ U = A ∪ B
Hence, A ∪ (B – A) = A ∪ B.
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