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 your answer.

Answer 1

Since A C A ∀ A ∈ P (X) R is reflexive. 

Also, for A, B, C ∈ P(X), A R B & B R C 

⇒ A C B and B C C 

⇒ A ⊂ C A R C 

∴ R is transitive. 

R is not symmetric as A C B 

⇒ B C A 

so ARB  BRA 

Hence R is not an equivalence relation.