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
\[\begin{align*} A \cup B &= B \cup A \\ A \cap B &= B \cap A \end{align*}\]b. Associativity
\[\begin{align*} A \cup (B \cup C) &= (A \cup B) \cup C \\ A \cap (B \cap C) &= (A \cap B) \cap C \end{align*}\]c. Distributive laws
\[\begin{align*} A \cap (B \cup C) &= (A \cap B) \cup (A \cap C) \\ A \cup (B \cap C) &= (A \cup B) \cap (A \cup C) \end{align*}\]d. DeMorgan’s Laws
\[\begin{align*} (A \cup B)^c &= A^c \cap B^c \\ (A \cap B)^c &= A^c \cup B^c \end{align*}\]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 $S$.