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 relation on $P(X)$? Justify your answer.

(D) $1$. Reflexivity: Since every set is a subset of itself,$A \subset A$ for all $A \in P(X)$. Therefore,$ARA$ holds for all $A \in P(X)$,so $R$ is reflexive.
$2$. Symmetry: For $R$ to be symmetric,$ARB$ must imply $BRA$. Here,$ARB$ means $A \subset B$. This does not imply $B \subset A$. For example,let $X = \{1, 2, 3\}$,$A = \{1\}$,and $B = \{1, 2\}$. Here $A \subset B$ ($ARB$ is true),but $B \not\subset A$ ($BRA$ is false). Thus,$R$ is not symmetric.
$3$. Transitivity: If $ARB$ and $BRC$,then $A \subset B$ and $B \subset C$. By the definition of subsets,this implies $A \subset C$,so $ARC$ holds. Thus,$R$ is transitive.
Conclusion: Since $R$ is not symmetric,it is not an equivalence relation.