Exercise 13.5 [exclusive-exhaustive-exercise]

This question deals with the properties of possible worlds, defined on page possible-worlds-page as assignments to all random variables. We will work with propositions that correspond to exactly one possible world because they pin down the assignments of all the variables. In probability theory, such propositions are called atomic event. For example, with Boolean variables $X_1$, $X_2$, $X_3$, the proposition $x_1\land \lnot x_2 \land \lnot x_3$ fixes the assignment of the variables; in the language of propositional logic, we would say it has exactly one model.

1. Prove, for the case of $n$ Boolean variables, that any two distinct atomic events are mutually exclusive; that is, their conjunction is equivalent to ${false}$.

2. Prove that the disjunction of all possible atomic events is logically equivalent to ${true}$.

3. Prove that any proposition is logically equivalent to the disjunction of the atomic events that entail its truth.