# The Sample Space and Set Operations

### The Sample Space

• The sample space, $\Omega$, is the set of all possible outcomes, $\omega$.
• Subsets of $\Omega$ are events.
• The empty set $\emptyset$ contains no elements.

#### Example – Tossing a coin

Toss a coin once: $\Omega = \{H, T\}$

Toss a coin twice: $\Omega = \{HH, HT, TH, TT\}$

Then event that the first toss is heads: $\omega = \{HH, HT\}$

### Set Operations – Complement, Union and Intersection

#### Complement

Given an event, $A$, the complement of $A$ is $A^\mathsf{c}$, where: $A^\mathsf{c} = \text{"Not A"} = \{\omega \in \Omega : \omega \notin A\}$

#### Union

The union of two sets A and B, $A \cup B$ is set of the events which are in either A, or in B or in both. $A \cup B = \{\omega \in \Omega : \omega \in A \text{ or } \omega \in B \text{ or } \omega \in both\}$ $\bigcup_{i=1}^{\infty} A_i = \{\omega \in \Omega : \omega \in A_i \text{ for at least one i} \}$

#### Intersection

The intersection of two sets A and B, $A \cap B$ is set of the events which are in both A and B. $A \cap B = \{\omega \in \Omega : \omega \in A \text{ and } \omega \in B\}$ $\bigcap_{i=1}^{\infty} A_i = \{\omega \in \Omega : \omega \in A_i \text{ for all i} \}$

#### Difference Set

The difference set is the events in one set which are not in the other: $A \setminus B = \{\omega : \omega \in A, \omega \notin B\}$

#### Subsets

If every element of A is contained in B then A is a subset of B: $A \subset B$ or equivalently, $B \supset A$.

#### Counting elements

If A is a finite set, then $|A|$ denotes the number of elements in A.

### Disjoint events

Two events A and B are disjoint or mutually exclusive if $A \cap B = \emptyset$ (the empty set) – i.e. there are no events in both A and B).
More generally, $A_1, A_2, ...$ are disjoint if $A_i \cap A_j = \emptyset$ whenever $i \neq j$.

#### Example – intervals of the real line

The intervals $A_1 = [0, 1), A_2 = [1, 2), A_3 = [2,3), ...$ are disjoint.
The intervals $A_1 = [0, 1], A_2 = [1, 2], A_3 = [2, 3], ...$ are not disjoint. For example, $A_1 \cap A_2 = \{1\}$.

### Partitions

A partition of the sample space $\Omega$ is a set of disjoint events $A_1, A_2, A_3, ...$ such that $\bigcup_{i=1}^{\infty} A_i = \Omega$.

### Monotone increasing and monotone decreasing sequences

A sequence of events, $A_1, A_2, ...$ is monotone increasing if $A_1 \subset A_2 \subset A_3 \subset ...$. Here we define $\lim_{n \to \infty} A_n = \bigcup_{i=1}^{\infty} A_i$ and write $A_n \to A$.

Similarly, a sequence of events, $A_1, A_2, ...$ is monotone decreasing if $A_1 \supset A_2 \supset A_3 \supset ...$. Here we define $\lim_{n \to \infty} A_n = \bigcap_{i=1}^{\infty} A_i$. Again we write $A_n \to A$