Hi everybody,

In VanDalen's "Logic and structure" he defines avaluationto be a function v:PROP->{0,1} such that for example v(phi ^ psi)=min(v(phi),v(psi))

and similar properties you know all, for other connectives.

then he discusses a theorem that lets us to extend an "atomic valuation"(A function h:ATOMS->{0,1})to a valuation on PROP.Indeed to have a unique valuation on PROP for any atomic valuation.

Later he asks to prove that [the number ofvaluationsis 2^(aleph-0)].

I have questions now:

1- Is ATOMS equinumerous to PROP?(Or it is only dominated by PROP?)

I am asking this beacause i want to know if they arenotequinumerous, then which "valuations" does he mean; Atomics valuations or The valuations on PROP?(Note that he asks the number of valuations just after he calls Atomic valuations also valuations)

2- What is Card(ATOMS) ? Because if we prove that ATOMS is equinumerous to natural numbers(i.e, has cardinality of (aleph-0))then we will have 2^aleph-0

(atomic)valuations.

Thanks.