Last updated at Sept. 3, 2021 by

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Example 31 For any sets A and B, show that P(A ∩ B) = P(A) ∩ P(B). To prove two sets equal, we need to prove that they are subset of each other i.e.. we have to prove P (A ∩ B) ⊂ P (A) ∩ P (B) & P(A) ∩ P(B) ⊂ P ( A ∩ B) Let a set X belong to Power set P(A ∩ B) i.e. X ∈ P ( A ∩ B ). As set X is in the power set of A ∩ B, X is a subset of A ∩ B because power set is the set of all subsets ⊂ Subset A ⊂ B (all elements of set A in set B) Thus, X is a subset of A ∩ B i.e. X ⊂ A ∩ B. So, X ⊂ A and X ⊂ B. Therefore, Since X is a subset of A & B, X is in power set of A and X is in power set of B i.e. X ∈ P(A) and X ∈ P(B) i.e. X ∈ P(A) and X ∈ P(B) ⇒ X ∈ P(A) ∩ P(B). So, if X ∈ P (A ∩ B), then X ∈ P(A) ∩ P(B) i.e. all elements of set P (A ∩ B) are in set P(A) ∩ P(B) Thus, gives P (A ∩ B) ⊂ P (A) ∩ P (B). Similarly, Let a set Y belong to Power set P(A) ∩ P(B) i.e. Y ∈ P (A) ∩ P(B). Then Y ∈ P (A) and Y ∈ P ( B ). As set Y is in the power set of A & B, Y is a subset of A & Y is a subset of B because power set is the set of all subsets Thus, Y ⊂ A and Y ⊂ B. ∴ Y ⊂ A ∩ B, Therefore, Since Y is a subset of A ∩ B, Y is in power set of A ∩ B ⇒ Y ∈ P ( A ∩ B ). So, if Y ∈ P (A) ∩ P(B) , then Y ∈ P ( A ∩ B ). This gives P (A) ∩ P (B) ⊂ P (A ∩ B) Now, Since P (A ∩ B) ⊂ P (A) ∩ P (B) & P(A) ∩ P(B) ⊂ P ( A ∩ B) Hence, P ( A ∩ B ) = P ( A ) ∩ P ( B ).

Examples

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Example 2 Important

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Example 5 Important

Example 6 (i)

Example 6 (ii) Important

Example 6 (iii)

Example 6 (iv)

Example 6 (v)

Example 7 Important

Example 8 (i)

Example 8 (ii)

Example 9 Important

Example 10

Example 11 Important

Example 12

Example 13 Important

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Example 18 Important Deleted for CBSE Board 2022 Exams

Example 19 Deleted for CBSE Board 2022 Exams

Example 20 Deleted for CBSE Board 2022 Exams

Example 21 Deleted for CBSE Board 2022 Exams

Example 22 Important Deleted for CBSE Board 2022 Exams

Example 23

Example 24 Important

Example 25 Important

Example 26

Example 27 Important

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Example 29 Important

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Example 31 You are here

Example 32 Important

Example 33 Important

Example 34 Important

Chapter 1 Class 11 Sets (Term 1)

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About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.