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NCERT Class XI Mathematics - Sets - Solutions

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Question : 60 of 73
Marks: +1, -0
Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.
Solution:  
Given that A ∩ B = A ∩ C and A ∪ B = A ∪ C
We know that A = A ∩ (A ∪ B) and A = A ∪ (A ∩ B)
∴ B = B ∪ (B ∩ A) = B ∪ (A ∩ B) = B ∪ (A ∩ C) [∴ A ∩ B = A ∩ C]
= (B ∪ A) ∩ (B ∪ C) [By distributive law]
= (A ∪ B) ∩ (B ∪ C) = (A ∪ C) ∩ (B ∪ C) [Since A ∪ B = A ∪ C]
= (C ∪ A) ∩ (C ∪ B)
= C ∪ (A ∩ B) [By distributive law]
= C ∪ (A ∩ C) [Since A ∩ B = A ∩ C]
= C ∪ (C ∩ A) = C.
Hence B = C.
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