In Pinter's "A Book of Abstract Algebra", Chapter 13 Exercise J1 asks the reader to prove that the relation $\sim$ on set $A$, where $\sim$ is defined as "$u \sim v\ $ iff $\ g(u)=v$ for some $g \in G$", is an equivalence relation.
To provide further context, $g \in G$ where $G$ is a subgroup of the symmetric group $S_A$. Further, $u,v \in A$.
I am comfortable with these styles of proofs (reflexivity, symmetry, transitivity), but would like to use this particular exercise to ask two questions about notation.
For the symmetry part of the proof, I need to prove the below implication:
If $\exists g \in G \backepsilon g(u)=v$, then $\exists h \in G \backepsilon h(v)=u$
This is straightforward and one realizes that $h$ is simply $g^{-1}$. However, I have a question regarding how to write the implication.
Specifically, does the following implication (where $h$ is replaced with $g$) denote the exact same thing?
If $\exists g \in G \backepsilon g(u)=v$, then $\exists g \in G \backepsilon g(v)=u$
Or does this force the $g$ in the antecedent to be the exact same $g$ that is in the consequent? (or does the value of $g$ from the antecedent "reset" when looking at the $g$ in the consequent)
With that question out of the way, I want to now ask how one would reframe the definition of $\sim$ such that one intentionally requires that the $g$ in the antecedent is the same as the $g$ in the consequent.
Said differently, what would I need to change in the definition of $\sim$ to make the "symmetry implication" read as follows:
If there is a specific element $g_1 \in G \backepsilon g_1(u)=v$, then for that same element $g_1$, $g_1(v)=u$