Set theory

The set, $S$ , of all possible outcomes of a particular experiment is called the sample space for the experiment.

An event is any collection of possible outcomes of an experiment, that is, any subset of $S$ (including $S$ itself).

Theorem: For any three events, $A$ , $B$ , and $C$ , defined on a sample space $S$ ,

a. Commutativity

\begin{aligned} A \cup B & = B \cup A \\ A \cap B & = B \cap A \end{aligned}

b. Associativity

\begin{aligned} A \cup (B \cup C) & = (A \cup B) \cup C \\ A \cap (B \cap C) & = (A \cap B) \cap C \end{aligned}

c. Distributive laws

\begin{aligned} A \cap (B \cup C) & = (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) & = (A \cup B) \cap (A \cup C) \end{aligned}

d. DeMorgan’s Laws

\begin{aligned} (A \cup B)^{c} & = A^{c} \cap B^{c} \\ (A \cap B)^{c} & = A^{c} \cup B^{c} \end{aligned}

Two events $A$ and $B$ are disjoint (or mutually exclusive) if $A \cap B = \emptyset$ . The events $A_{1}, A_{2}, \dots$ are pairwise disjoint if $A_{i} \cap A_{j} = \emptyset$ for all $i \neq j$ .

If $A_{1}, A_{2}, \dots$ are pairwise disjoint and $\cup_{i = 1}^{\infty} A_{i} = S$ , then the collection $A_{1}, A_{2}, \dots$ forms a partition of $S$ .