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.
Set Operations
Suppose we have four sets  W, X, Y, and Z. Let these sets be defined as
follows: W = {2}; X = {1, 2}; Y= {2, 3, 4}; and Z = {1, 2, 3, 4}.

The union
of two sets is the set of elements that belong to one or both of the two sets.
Thus, set Z is the union of sets X and Y.

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, set W is
the intersection of sets X and Y.

Symbolically, the intersection of X and Y is denoted by X
∩ Y.
Sample Problems

Describe the set of vowels.
If A is the set of vowels, then A could be described as A = {a, e, i, o,
u}.

Describe the set of positive integers.
Since it would be impossible to list all of the positive integers, we
need to use a rule to describe this set. We might say A consists of all
integers greater than zero.

Set A = {1, 2, 3} and Set B = {3, 2, 1}. Is Set A equal to
Set B?
Yes. Two sets are equal if they have the same elements. The order in which the
elements are listed does not matter.

What is the set of men with four arms?
Since all men have two arms at most, the set of men with four arms contains no
elements. It is the null set (or empty set).

Set A = {1, 2, 3} and Set B = {1, 2, 4, 5, 6}. Is Set A a
subset of Set B?
Set A would be a subset of Set B if every element from Set A
were also in Set B. However, this is not the case. The number 3 is in
Set A, but not in Set B. Therefore, Set A is not a subset
of Set B.