Properties of the inclusion and membership relations

Statement

A relation $R$ is reflexive if $\forall x\,(x R x)$ and symmetric if $\forall x \forall y\,(x R y \to y R x)$. Show that $\subseteq$ is reflexive but $\in$ is not. Then decide whether $\subseteq$ and $\in$ are symmetric.

Solution

$\subseteq$ is reflexive: every set is a subset of itself, $A \subseteq A$.

$\in$ is not reflexive: in general $A \notin A$. For instance $\{1\} \notin \{1\}$, because the only element of $\{1\}$ is $1$, not $\{1\}$. This counterexample suffices.

$\subseteq$ is not symmetric: $A \subseteq B$ does not imply $B \subseteq A$. Counterexample: $\{1\} \subseteq \{1,2\}$ but $\{1,2\} \not\subseteq \{1\}$.

$\in$ is not symmetric: $A \in B$ does not imply $B \in A$. Counterexample: $1 \in \{1\}$ but $\{1\} \notin 1$. $\blacksquare$