Misc 3 - Let A U B = A U C and AB = AC. show that B = C.

Misc 3 - Chapter 1 Class 11 Sets - Part 2
Misc 3 - Chapter 1 Class 11 Sets - Part 3 Misc 3 - Chapter 1 Class 11 Sets - Part 4

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Misc 3 Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C. In order to prove B = C, we should prove B is a subset of C i.e. B ⊂ C & C is a subset of B i.e. C ⊂ B Let x ∈ B ⇒ x ∈ A ∪ B ⇒ x ∈ A ∪ C ⇒ x ∈ A or x ∈ C (Since B ⊂ A ∪ B, all elements of B are in A ∪ B) (Given A ∪ B = A ∪ C) Taking x ∈ A x ∈ A Also, x ∈ B ∴ x ∈ A ∩ B (If x belongs to both A and B ,it will belong to common of A & B also) x ∈ A ∩ B So, x ∈ A ∩ C i.e. x ∈ A and x ∈ C i.e. x ∈ C ∴ If x ∈ B , then x ∈ C i.e. if an elements belongs to set B, then it must belong to set C also ∴ B ⊂ C (From (1)) (Given A ∩ B = A ∩ C) Taking x ∈ C x ∈ C Also, x ∈ B ∴ If x ∈ B , then x ∈ C i.e. if an elements belongs to set B, then it must belong to set C also ∴ B ⊂ C Similarly, we can prove C ⊂ B From (3) and (4) B ⊂ C and C ⊂ B ⇒ B = C (From (1)) Since B ⊂ C and C ⊂ B ⇒ B = C Hence proved

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.