Given a non-empty set X, define the relation R in P(X) as follows: For A, B ∈ ð(ð), (ðī, ðĩ) ∈ ð iff ðī ⊂ ðĩ. Prove that R is reflexive, transitive and not symmetric.
CBSE Class 12 Sample Paper for 2023 Boards
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CBSE Class 12 Sample Paper for 2023 Boards
Last updated at April 16, 2024 by Teachoo
Question 33 (Choice 2) - Introduction Given a non-empty set X, define the relation R in P(X) as follows: For A, B â ð(ð), (ðī, ðĩ) â ð iff ðī â ðĩ. 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 = { ð, {1} , {2} , {3}, {1, 2} , {2, 3} , {1, 3}, {1, 2, 3} } Since {1} â {1, 2} âī {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 â ð(ð), (ðī, ðĩ) â ð iff ðī â ðĩ. Prove that R is reflexive, transitive and not symmetric.ARB means A â B Here, relation is R = {(A, B): A & B are sets, A â B} Check reflexive Since every set is a subset of itself, A â A âī (A, A) â R. âīR is reflexive. Check symmetric To check whether symmetric or not, If (A, B) â R, then (B, A) â R If (A, B) â R, A â B. But, B â A is not true Example: Let A = {1} and B = {1, 2}, As all elements of A are in B, A â B But all elements of B are not in A (as 2 is not in A), So B â A is not true âī R is not symmetric. Checking transitive Since (A, B) â R & (B, C) â R If, A â B and B â C. then A â C â (A, C) â R So, If (A, B) â R & (B, C) â R , then (A, C) â R âī R is transitive. Hence, R is reflexive and transitive but not symmetric.