I am trying to learn math on my own and am using Frédéric Mynard's "An Introduction to the language of Mathematics" as an introductory textbook for logic and proofs. Unfortunately, most of the exercises don't contain solutions, and I ignore if I am right or wrong with the answers. Could you provide advice about whether my answer for the following exercise is correct?
If the universe of discourse is the set of players, introduce appropriate predicates and propositions to give the logical form of the statement “The players will go back to work if agreement is reached about their salaries, but this will be achieved, if at all, only if some of them take early retirement.”
Let A(x): “x reaches an agreement about his salary”, G(x): “x goes back to work” and ER(y): “y takes early retirement”. Then:
∀x ((A(x) → G(x)) ∧ [((A(x) → G(x)) → (∃y ER(y))]
Thanks in advance for your time!