My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this:
Let $X \subseteq R^2$. A symmetry of $X$ is an isometry $f: R^2 \to R^2$ such that, $ \forall x \in X, f(x) \in X$
Are they the same? If not, why? The problem is that being the domain of the function the points of the plane, you can only plug into it points of the plane, not sets of points. So $ f(X)$ and $f(X)=X$ should be defined first, and the only definition I can think of is the one I've given above, but I would like to be corrected.
They differ. Let $X=\{(a,b)\in\mathbb{R}^2\mid a\geq 0\}$. Then $f(a,b)=(a+1,b)$ is a symmetry by your definition but not by the definition $f(X)=X$ because $f$ is not surjective onto $X$ in this case.