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 Ì B. Is R an equivalence relation on P (X) ? Justify your answer

Solution:

Since every set is a subset of itself,

ARA for all A ∈ P (X).

⇒ R is reflexive.

Let ARB ⇒ A ⊂ B

This cannot be implied to B ⊂ A.

If A = {1, 2} and B = {1, 2, 3},

then it cannot be implied that B is related to A.

Therefore,

R is not symmetric.

If ARB and BRC , then A ⊂ B and B ⊂ C.

⇒ A ⊂ C

⇒ ARC

⇒ R is transitive.

R is not an equivalence relation as it is not symmetric

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise ME Question 8

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 Ì B. Is R an equivalence relation on P (X) ? Justify your answer

Summary:

R is transitive. R is reflexive.R is not symmetric. R is not an equivalence relation as it is not symmetric