Set Theory

Study guide: Exam P-1

Set Theory is the branch of mathematics that studies collections of objects called sets and the relationships between them. It supplies the language of Probability: the Sample Space $S$ is the set of all outcomes, an Event is a subset of $S$ , and Set Operations on events (union, intersection, complement) mirror the words "or", "and", and "not".

$A \subseteq S \iff (\omega \in A \Rightarrow \omega \in S)$

A set is specified by listing its elements, $A = \{1, 2, 3\}$ , or by a defining property, $A = \{\, \omega : \omega \text{ satisfies } P \,\}$ .
$\omega \in A$ means $\omega$ is an element of $A$ ; $A \subseteq S$ means every element of $A$ also lies in $S$ .
The empty set $\varnothing$ contains no elements; the complement $A^c = S \setminus A$ is everything in $S$ not in $A$ .
Sets are the building blocks of a probability space $(S, \mathcal{F}, P)$ , in which events are the sets whose probabilities are measured.

Union, Intersection, and Difference of Two Sets Given $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$ , find $A \cup B$ , $A \cap B$ , and $A \setminus B$ .

Answer $A \cup B = \{1, 2, 3, 4, 5, 6\}$ $A \cap B = \{3, 4\}$ $A \setminus B = \{1, 2\}$ The union collects elements in either set, the intersection those in both, and the difference those in $A$ only.

Complement and De Morgan's Law A Sample Space for a die roll is $S = \{1, 2, 3, 4, 5, 6\}$ . Let $A = \{2, 4, 6\}$ (even) and $B = \{4, 5, 6\}$ (greater than 3). Find $A^c$ and verify $(A \cup B)^c = A^c \cap B^c$ .

Answer The complements are $A^c = S \setminus A = \{1, 3, 5\}$ and $B^c = \{1, 2, 3\}$ . Then: $A \cup B = \{2, 4, 5, 6\} \implies (A \cup B)^c = \{1, 3\}$ $A^c \cap B^c = \{1, 3, 5\} \cap \{1, 2, 3\} = \{1, 3\}$ Both sides equal $\{1, 3\}$ , confirming De Morgan's Law.