State the Completeness Theorem for Propositional Logic.

[You do not need to give definitions of the various terms involved.]

State the Compactness Theorem and the Decidability Theorem, and deduce them from the Completeness Theorem.

A set $S$ of propositions is called finitary if there exists a finite set $T$ of propositions such that $\{t: S \vdash t\}=\{t: T \vdash t\}$. Give examples to show that an infinite set of propositions may or may not be finitary.

Now let $A$ and $B$ be sets of propositions such that every valuation is a model of exactly one of $A$ and $B$. Show that there exist finite subsets $A^{\prime}$ of $A$ and $B^{\prime}$ of $B$ with $A^{\prime} \cup B^{\prime} \models \perp$, and deduce that $A$ and $B$ are finitary.