4. (25 points) This question deals with factored representations, which we discussed in class are representations of possible worlds denoted by a set of assignments to a collection of 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 events. For example, with Boolean variables,
X 1
,X 2
,X 3
, the proposition
x 1
∧¬x 2
∧¬x 3
fixes the assignment of the variables. In the language of propositional logic (remember CSE 2500?) we would say it has exactly one model. Prove by induction that for the case of
n
Boolean variables, that any two distinct atomic events are mutually exclusive; that is, their conjunction is equivalent to false.
