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 $A$ and $B$ in $P(X),ARB$, if and only if $A\subset B$. Then, $R$ is

• A. reflexive
• B. transitive
• C. not symmetric
• D. an equivalence relation

Answer 1

Answer: (C)

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 $B\subset C$.
$A\subset C\Rightarrow ARC\text{. }$
Therefore, $R$ is transitive.
Hence, $R$ is not an equivalence relation, since it is not symmetric.