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Example 30 - Show that A U B = AB implies A = B - Chapter 1 Sets - Proof - where properties of sets cant be applied,using element

  1. Chapter 1 Class 11 Sets
  2. Serial order wise
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Example 30 Show that A ∪ B = A ∩ B implies A = B 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 ∈ A. Then x ∈ A ∪ B. Since A ∪ B = A ∩ B , ⇒ x ∈ A ∩ B. So x ∈ B. ∴ If x ∈ A , then x ∈ B i.e. if an elements belongs to set A, then it must belong to set B also Therefore, A ⊂ B. Similarly, if y ∈ B, then y ∈ A ∪ B. Since A ∪ B = A ∩ B, y ∈ A ∩ B. So, y ∈ A. ∴ If y ∈ B , then y ∈ A i.e. if an elements belongs to set B, then it must belong to set A also Therefore, B ⊂ A. From (1) & (2) A ⊂ B & B ⊂ A Thus, A = B Hence shown

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Davneet Singh
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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