Sets and Subsets
The lesson introduces the important topic of sets, a simple idea that recurs
throughout the study of probability and statistics.
Set Definitions

A set
is a welldefined collection of objects.

Each object in a set is called an element
of the set.

Two sets are equal
if they have exactly the same elements in them.

A set that contains no elements is called a null set or an
empty set.

If every element in Set A is also in Set B, then Set A is
a subset of Set B.
Set Notation

A set is usually denoted by a capital letter, such as A, B, or C.

An element of a set is usually denoted by a small letter, such as x, y, or
z.

A set may be described by listing all of its elements enclosed in braces. For
example, if Set A consists of the numbers 2, 4, 6, and 8, we may say: A
= {2, 4, 6, 8}.

The null set is denoted by
{} or ∅.

Sets may also be described by stating a rule. We could describe Set A from
the previous example by stating: Set A consists of all the even
singledigit positive integers.
Sets and Probability
As we learned in the previous lesson,
probability is all about statistical experiments.
When a researcher conducts a statistical experiment,
he or she cannot know the outcome in advance. The outcome is determined by chance.
However, if the researcher can list all the possible outcomes of the experiment, it may be possible to compute the probability of a particular outcome.
The list of all possible outcomes from a statistical experiment is called the sample space. And a particular outcome or
collection of outcomes is called an event.
You can see that a sample space is a type of set. It is a welldefined listing of all possible outcomes from a statistical experiment. And an
event in a statistical experiment is a subset of the sample space.
Set Operations
Suppose we have a sample space S defined as follows: S = {1, 2, 3, 4, 5, 6}.
Within that sample space, suppose we define two subsets as
follows: X = {1, 2} and Y= {2, 3, 4}.

The union
of two sets is the set of elements that belong to one or both of the two sets.
Thus, if X is {1, 2} and Y is {2, 3, 4}, the union of sets X and Y is:
X ∪ Y = {1, 2, 3, 4}
Symbolically, the union of X and Y is denoted by X
∪ Y.

The intersection
of two sets is the set of elements that are common to both sets. Thus, if X is {1, 2} and Y is {2, 3, 4}, the intersection of sets X and Y is:
X ∩ Y = {2}
Symbolically, the intersection of X and Y is denoted by X
∩ Y.

The complement
of an event is the set of all elements in the sample space but not in the event.
Thus, if the sample space is {1, 2, 3, 4, 5, 6}, and Y is {2, 3, 4}, the complement of set Y is:
Y' = {1, 5, 6}
On this website, we denote the complement of set Y as Y'. In other places, you may see the complement of set Y denoted as Y^{c}.