Given a non empty set & 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 it A ⊂B.  Then R is not

Since every set is a subset of itself, ARA for all A∈P(X)Therefore, .R is reflexive.  Let ARB⇒A⊂B .  This cannot be implied to B⊂A .  For instant, 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.  Further if ARB and BRC , then A⊂B and B⊂C . ⇒ A⊂C⇒ ARCTherefore, R is transitive. Hence, R is not an equivalence relation since it is not symmetric