3.1 Types of Sets and Set Notation
The following are definitions that will be used throughout the first half of this unit:
Set
A collection of distinguishable objects; for example, the set of whole numbers is:
W = {0, 1, 2, 3, …}
A set can be written in one of three ways:
 listing every element in the set: A = {6, 7, 8, 9}
 using words or a sentence: A = {all integers greater than 5 and less than 10}
 using set notation: {𝑥|5 < 𝑥 < 10, 𝑥 ∈ 𝐼}
Element
An object in a set; for example, 3 is an element of D = {single digit numbers}
Number of Elements in a Set
The number of elements in a set is given using the following notation: 𝑛(𝐴). For example, 𝑛(𝐷) (using
set D above) would be equal to 10.
Universal Set
A set of all the elements under consideration for a particular context (also called the sample space); for
example, the universal set of single digit numbers is:
D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
Subset
A set whose elements all belong to another set: for example, the set of odd digits,
O = {1, 3, 5, 7, 9}, is a subset of D, the set of digits. In set notation, this relationship is written as O ⊂ D
Complement
All the elements of a universal set that do not belong to a subset of it; for example,
O’ = {0, 2, 4, 6, 8} is the complement of O = {1, 3, 5, 7, 9}, a subset of the universal set of digits, D. The
complement is denoted with a prime sign, O’.
Empty Set
A set with no elements, for example, the set of odd numbers divisible by 2 is the empty set.
The empty set is denoted by { } or ∅.
Disjoint
Two or more sets that have NO elements in common; for example, E = {even numbers} and
O = {odd numbers} are disjoint.
Example 1
1. List the elements of the universal set of Canadian provinces and territories, C.
C={
n(C) =
2. One subset of C is the set of Western Provinces and territories, W. Write W below.
W={
n(W) =
3. The complement of W is the set W’. Write W’ below.
W’ = {
n(W’) =
4. A student wrote the set of Eastern provinces as follows: E = {NL, PE, NS, NB, QC, ON}
n(E) =
Is E equal to W’? Explain
5. List T, the set of territories in Canada. Is T a subset of C? Is it a subset of W, or a subset of W’?
T={
6. Create a Venn diagram to show C, W, E, and T.
7. List the set of Canadian provinces south of Mexico (call this set S).
S={
8. Consider the sets C, W, E, and T. List a pair of disjoint sets.
Example 2
Alden and Connie rescue homeless animals. They are setting up a web page to help them advertise the
animals that are available for adoption. They currently have dogs, cats, rabbits, ferrets, parrots,
lovebirds, macaws, iguanas, and snakes. You are going to help them come up with a logical scheme by
which the web page will be organized.
a) Define and write the following sets: The universal set (A), the set of warm-blooded animals (W),
and the set of cold-blooded animals (C).
b) Organize the subset of warm-blooded animals into two further subsets: mammals (M) and
birds (B).
c) Represent your organization scheme using a Venn Diagram. List each element in its appropriate
place in the Venn Diagram.
d) Name any/all pairs of disjoint sets.
e) Show which sets are subsets of one another using set notation.
Example 3
a) Write the following sets (use the given letters for the names of the sets):

all integers from 1 to 20 (U) = {

all multiples of 5 from 1 to 20 (F) = {

all multiples of 10 from 1 to 20 (T) = {

all multiples of 3 from 1 to 20 (A) = {

all multiples of 7 from 1 to 20 (S) = {
b) Draw a Venn diagram to represent the sets you wrote above.
c) List the disjoint subsets, if there are any.
d) Is each of the following statements true or false? Explain.
i) 𝐹 ⊂ 𝑇
ii) 𝑇 ⊂ 𝐹
iii) 𝑆 ⊂ 𝑆
iv) F’ = {multiples of 10 from 1 to 20}
v) For this example, the set of integers from 21 to 30 is { }.