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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.
Solution
since every set is a subset of itself, $ARA $ for all $A \in P ( X )$
$\therefore R$ is reflexive.
Let $ARB \Rightarrow A \subset B$
This cannot be implied to $B \subset A$.
For instance, if $A =\{1,2\}$ and $B =\{1,2,3\},$ then it cannot be implied that $B$ is related to $A$.
$\therefore R$ is not symmetric.
Further, if $ARB$ and $BRC$, then $A \subset B$ and $B \subset C$.
$\Rightarrow A \subset C$
$\Rightarrow ARC$
$\therefore R$ is transitive.
Hence, $R$ is not an equivalence relation as it is not symmetric.
