Check sibling questions

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

 

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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), (ðī, ðĩ) ∈ 𝑅 iff ðī ⊂ ðĩ. 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} âˆī {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), (ðī, ðĩ) ∈ 𝑅 iff ðī ⊂ ðĩ. 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 âˆī (A, A) ∈ R. âˆīR 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 âˆī 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 âˆī 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.