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Misc 3 (Introduction) Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify you answer: 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} If A ⊂ B, all elements of A are in B Misc 3 Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the relation R in P(X) as follows: For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify you answer: 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. If A ⊂ B, all elements of A are in B 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. Hence, R is not an equivalence relation since it is not symmetric.

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Davneet Singh

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.