I was given a definition.
A symmetry of a plane figure $F$ is an isometry that maps $F$ to itself, that is, an isometry $f:R^2 \to R^2$ such that $f(F)=F$.
I don't really understand this because is $F$ not a collection of points and the domain of the function consists of single points in $R^2$?
Can someone help me out? Thanks.
Yes $F$ is a subset of $R^2$.
$f(F)$ is shorthand for the set $\{f(x) : x \in F\}$. The statement "$f(F) = F$" is a set equality.