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$\subset$B. Is R an equivalence relations on P(X) ? Justify your answer.

(i) Since A $\subset$A $\forall A \in$ P (X ) $\Rightarrow$ ARA

$\therefore$ R is reflexive.

(ii) Let ARB $\Rightarrow$ A $\subset$B and BRA $\Rightarrow$ B $\subset$ A

$\Rightarrow$ A $=$ B (which is not so) [
$\Rightarrow$ ARB $\not \Rightarrow$BRA $\Rightarrow$ R is not symmetric

(iii) ARB,BRC $\Rightarrow$ A$\subset$B, B$\subset$ C $\Rightarrow$ f $\subset$C $\Rightarrow$ ARC
$\Rightarrow$ R is transitive

Thus R is not an equivalence relation of P(X).