Difference between "Let x be a ..." and "Let x ... be arbitrary"

Question

What is the difference between "Let x be a ..." and "Let x ... be arbitrary"?

Consider the following example:

• Let $R$ be an equivalence Relation on $A$. Then $\forall x, y\in A.\ [x]_R=[y]_R.$
• Let $R$ be an arbitrary equivalence Relation on $A$. Then $\forall x, y\in A.\ [x]_R=[y]_R.$

Answer 1

There is no difference. "Let x be an ... " implies that it's any/arbitrary.
"Let x be an arbitrary ... " just makes this implication more explicit.
But formally speaking, they mean the same.