TITLE: Give a relation $X\subseteq A\times A$, so that $R\subseteq X$ and $X$ is symmetric
QUESTION [0 upvotes]: $R=\{(2,5),(3,4)\}$
$A=\{2,3,4,5\}$
my answer:
$X=\{(2,5)(5,2)\}$
$X$ is a proper subset because it contains not all elements of $A\times A$ and not all elements of $R$. And $X$ is symmetric because it contains the pairs $(2,5)$ and $(5,2)$
Is this assumption correct?

REPLY [2 votes]: No, since $R$ should be a subset of $X$. Instead, from your answer we have that $(3,4) \in R$ but $(3,4) \notin X$. So to fix this, we should put $(3,4)$ in $X$. But then since $X$ must be symmetric, what else must also go in $X$?
Note that $R$ is a proper subset of $X$ iff every element of $R$ is also in $X$ but $X$ contains an element that is not in $R$. Since $(5,2) \in X$ but $(5,2) \notin R$, we don't need to worry about $X$ being improper.