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 May 29, 2023 by Teachoo
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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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.
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.
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