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.