Specturm 3b in a link

Question

Can someone help with #3b in this link:

http://homepages.math.uic.edu/~marker/math502f09/ps3.pdf

I am trying to practice idea of spectrum, but cannot quite understand whatt they are asking for.

Thanks

Answer 1

Well, according to the definition of "spectrum" at the beginning of Problem 3, they are asking you to prove that the sentence $\phi$ in 3b (the conjunction of the four displayed sentences) has a model of size $n$ (a natural number) if and only if $n>3$ and $n$ is not prime. If you don't "see" it easily, I suggest trying to build models of sizes $4$, $5$, and $6$ (i.e., take sets of those three sizes and try to interpret $P$, $Q$, and $f$ in them so as to make the four displayed sentences true). Once you see why you can succeed for $4$ and $6$ but not for $5$, the general situation should become clear. (It might also be useful to think first about the last two of the four displayed sentences; although they're longer and more complicated than the first two, they incorporate the crucial idea.)