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.

Question

Philoid · Accepted Answer

We know that every set is a subset of itself, ARA for all A ϵ P(X).

⇒ R is reflexive.

This cannot be implied to B ⊂ A.

So, if A = {1, 2} and B = {1, 2, 3}, then it cannot be implied that B is related to A.

⇒ R is not symmetric.

So, if ARB and BRC, then A ⊂ B and B ⊂ C.

⇒ A ⊂ C

⇒ R is transitive.

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

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.