You are given the signagure $σ = 〈 ; ; P〉$ in which P is a unary predicate symbol. Find a formula F such that $\forall x F = 0$ and $\exists x$ $F = 1$. Prove that it meets these requirements.
From what I understand, we need to assume that the domain of discourse D may be any non-empty set. Also we need to assume that the predicate P may be either true for all $x \in D$ (1), for some $x \in D$ (2) or for no $x \in D$ (3).
I tried to solve it in 3 parts. In case (1) the appropriate formula F would just be P(x). But then once I tried to do the other cases, I got stuck since if all of your values are 0 or 1, how can you change only some of them around by only using the symbols we have $(\neg, \land, \lor, \rightarrow, \leftrightarrow, \forall, \exists)$? Therefore, help would be much appreciated.