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
\[\as{ A \cup B &= B \cup A \\ A \cap B &= B \cap A }\]b. Associativity
\[\as{ A \cup (B \cup C) &= (A \cup B) \cup C \\ A \cap (B \cap C) &= (A \cap B) \cap C }\]c. Distributive laws
\[\as{ A \cap (B \cup C) &= (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) &= (A \cup B) \cap (A \cup C) }\]d. DeMorgan’s Laws
\[\as{ (A \cup B)^c &= A^c \cap B^c \\ (A \cap B)^c &= A^c \cup B^c }\]Two events \(A\) and \(B\) are disjoint (or mutually exclusive) if \(A \cap B=\emptyset\). The events \(A_1, A_2, \ldots\) are pairwise disjoint if \(\A_i \cap \A_j=\emptyset\) for all \(i \ne j\).
If \(A_1,A_2,\ldots\) are pairwise disjoint and \(\cup_{i=1}^\infty A_i=S\), then the collection \(A_1,A_2,\ldots\) forms a partition of \(S\).