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

  1. Chapter 1 Class 11 Sets
  2. Serial order wise
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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. Inorder to prove A=B, we should prove A is a subset of B i.e. A ⊂ B & B is a subset of A i.e. B ⊂ A Let x ∈ B ⇒ x ∈ A ∪ B ⇒ x ∈ A ∪ C ⇒ x ∈ A or x ∈ C (2) Taking x ∈ A x ∈ A (2) Also, x ∈ B ∴ x ∈ A ∩ B (if x belongs to both A and B ,it will belong to common of A and B also) ⇒ x ∈ A ∩ C ∴ x ∈ A and x ∈ C ∴ x ∈ C 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 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 Since B ⊂ C and C ⊂ B ⇒ B = C Hence proved

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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