A predicate logic question about write down a sentence

Question

Let $\mathcal{L}=\{f\}$ be a first-order language containing a unary function symbol f, and no other non-logical symbols. Write down sentences $φ$ and $ψ$ of $\mathcal{L}$ such that for any $\mathcal{L}$-structure $\mathcal{A}=<A, f_{\mathcal{a}}>$

Write down a sentence $ρ$ such that whenever $\mathcal{A}\models ρ$ and $A$ is finite, then $A$ contains an even number of elements and, further, every finite set with an even number of elements is the domain of some model of $ρ$. What can you say about the size of the domains of the models of the sentence $¬ρ$?

Hi, can someone give me a hint about what kind of unary function can make it have even elements. I tried square root, when domain contains 0, it always cannot work out. :(

also, I dont really understand the question about "what can you say about the size of the domains..." what does it ask for?

Answer 1

A finite set has even size exactly if its elements can be paired up two by two.

A natural way to express this pairing when you have only a function to express it with would be to require that the function should map each element to its partner.

Can you write down the conditions "$f$ maps every element to something that maps back to the element we started with" and "$f$ doesn't map anything to itself" as $\mathcal L$-sentences?

I tried square root, when domain contains 0, it always cannot work out. :(

That is definitely a wrong direction. You don't get to specify what $f$ must be, only the sentence $\rho$ that it will satisfy -- and then it must work out such that every possible $f$ that satisfies $\rho$ has a finite domain has a domain of even size.