**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.

This question deals with the properties
of possible worlds, defined on pageĀ 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.