Question

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:

Answer 1

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. 
For instance, if A = {1, 2} and B = {1, 2, 3}, 
then it cannot be implied that B is related to A. 
∴ R is not symmetric. 
Further, if ARB and BRC, 
then A ⊂ B and B ⊂ C. 
\(\Rightarrow\) A ⊂ C \(\Rightarrow\) ARC 
∴ R is transitive. 

Hence, R is not an equivalence relation since it is not symmetric.