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Question 33 (Choice 2) - CBSE Class 12 Sample Paper for 2023 Boards - Solutions of Sample Papers and Past Year Papers - for Class 12 Boards

Last updated at April 16, 2024 by

Given a non-empty set X, define the relation R in P(X) as follows: For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 iff 𝐴 ⊂ 𝐵. Prove that R is reflexive, transitive and not symmetric.

 

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Transcript

Question 33 (Choice 2) - Introduction Given a non-empty set X, define the relation R in P(X) as follows: For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 iff 𝐴 ⊂ 𝐵. Prove that R is reflexive, transitive and not symmetric.Taking an example Let X = {1, 2, 3} P(X) = Power set of X = Set of all subsets of X = { 𝜙, {1} , {2} , {3}, {1, 2} , {2, 3} , {1, 3}, {1, 2, 3} } Since {1} ⊂ {1, 2} ∴ {1} R {1, 2} Question 33 (Choice 2) Given a non-empty set X, define the relation R in P(X) as follows: For A, B ∈ 𝑃(𝑋), (𝐴, 𝐵) ∈ 𝑅 iff 𝐴 ⊂ 𝐵. Prove that R is reflexive, transitive and not symmetric.ARB means A ⊂ B Here, relation is R = {(A, B): A & B are sets, A ⊂ B} Check reflexive Since every set is a subset of itself, A ⊂ A ∴ (A, A) ∈ R. ∴R is reflexive. Check symmetric To check whether symmetric or not, If (A, B) ∈ R, then (B, A) ∈ R If (A, B) ∈ R, A ⊂ B. But, B ⊂ A is not true Example: Let A = {1} and B = {1, 2}, As all elements of A are in B, A ⊂ B But all elements of B are not in A (as 2 is not in A), So B ⊂ A is not true ∴ R is not symmetric. Checking transitive Since (A, B) ∈ R & (B, C) ∈ R If, A ⊂ B and B ⊂ C. then A ⊂ C ⇒ (A, C) ∈ R So, If (A, B) ∈ R & (B, C) ∈ R , then (A, C) ∈ R ∴ R is transitive. Hence, R is reflexive and transitive but not symmetric.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.