What does it mean for a set to be isometric with another set?

Question

If a set $A$ is isometric with a set $B$, does that mean there exists an isometry from $A$ to $B$? Or is it the other way around, that there exists an isometry from $B$ to $A$?

Edit: I forgot that there were multiple definitions of an isometry, but my book's definition does not require it to be surjective. Sorry about that.

Answer 1

The answer is yes to both questions. Any isometry is a bijection onto its range, and the inverse is also a bijection. This means that there exists an isometry such that $f(A)=B$ if and only if there exists an isometry $g$ such that $g(B)=A$.

Answer 2

It means that there is a surjective isometry from $A$ to $B$ and it also means that there as a surjective isometry from $B$ to $A$. These assertions are equivalent.