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 $P(X), A R B$, 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, $A R A$ for all $A \in P(X)$. Therefore, $R$ is reflexive.
Let $ A R B \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 $A R B$ and $B R C$, then $A \subset B$ and $B \subset C$.
$A \subset C \Rightarrow A R C \text {. }$
Therefore, $R$ is transitive.
Hence, $R$ is not an equivalence relation, since it is not symmetric.