Question

Given a non-empty set X, consider P(X) which is the set of all such sets 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

(i) A⊂A ⇒ R is reflexive.

(ii) A ⊂ B, B ⊂ A ∴ R is not symmetric.

(iii) If A ⊂ B, B ⊂ C, then A ⊂ C ∴ R is transitive.

⇒ R is not an equivalence relation.