Let $P$ be a set of primitive propositions. Let $L(P)$ denote the set of all compound propositions over $P$, and let $S$ be a subset of $L(P)$. Consider the relation $\preceq$ on $L(P)$ defined by

$$s \preceq s t \text { if and only if } S \cup\{s\} \vdash t .$$

Prove that $\preceq_{S}$ is reflexive and transitive. Deduce that if we define $\approx_{S}$ by $\left(s \approx_{S} t\right.$ if and only if $s \preceq_{S} t$ and $\left.t \preceq \preceq_{S} s\right)$, then $\approx_{S}$ is an equivalence relation and the quotient $B_{S}=L(P) / \approx_{S}$ is partially ordered by the relation $\leqslant_{S}$ induced by $\preccurlyeq_{S}$ (that is, $[s] \leqslant_{S}[t]$ if and only if $s \preccurlyeq_{S} t$, where square brackets denote equivalence classes).

Assuming the result that $B_{S}$ is a Boolean algebra with lattice operations induced by the logical operations on $L(P)$ (that is, $[s] \wedge[t]=[s \wedge t]$, etc.), show that there is a bijection between the following two sets:

(a) The set of lattice homomorphisms $B_{S} \rightarrow\{0,1\}$.

(b) The set of models of the propositional theory $S$.

Deduce that the completeness theorem for propositional logic is equivalent to the assertion that, for any Boolean algebra $B$ with more than one element, there exists a homomorphism $B \rightarrow\{0,1\}$.

[You may assume the result that the completeness theorem implies the compactness theorem.]