What I already knew: Normally, I understand that "→" means implication, whereas "∧" means conjunction. The answer to this question is:
c. P(-5)∧P(-3)∧P(-1)∧P(3)∧P(5)
d. P(1)∨P(3)∨P(5)
It seems that "∀" and "∃" tells me whether to use "∧" or "∨", but I don't understand what the ∧ and → do?
If you find my question to be too confusing, could you solve this: What happens if I replaced → with ∧ in question C, or ∧ with → in question D?
I have to ask this question again despite it already exists because the original question asks about translation, but I still can't understand enough to see the difference in this context. I'm still learning.
The basic difference between $\to$ and $\land$ is that while $(−5≠1)→P(−5)$ is True because the antecedent is False, we have that $(−5≠1) \land P(−5)$ is False because at least the first conjunct is False.
Regarding the problem above, we have that in a finite domain $∀$ acts as a conjunction while $∃$ as a disjunction. Thus, d) amounts to $[(−5≥0) ∧ P(−5)] \lor \ldots \lor [(3≥0) ∧ P(3)]$ and c) to $[(−5 \ne 1) → P(−5)] ∧ \ldots ∧ [(3 \ne 1) → P(3)]$.
See also Quantifier (logic) and Bounded quantifier: $(∃n<t) \varphi$ means $∃n(n<t ∧ \varphi)$ while $(∀n<t) \varphi$ means $∀n(n<t → \varphi)$.