Let L = {f } be a ﬁrst-order language containing a unary function

symbol f , and no other non-logical symbols.

1.Write down a sentence χ of L which is satisﬁable in some structure

with an inﬁnite domain but is false in every structure with a ﬁnite domain.

What can you say about the size of the domains of the models of the sentence

2.Write down a sentence ρ such that whenever A |= 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?

Could anyone please give me some hints how to deal with this problem? I'm not sure where to start. Any help is appreciated!