Section 1.1 - An Introduction to Sets
Definitions:

Set - a set is a collection of objects.

Element - Each object in the set is called an element or member of the set.
Two Special Sets:
1) Natural numbers - The set of counting numbers 1, 2, 3, 4, 5, … is called the set of Natural numbers.
The set of natural numbers is denoted N.
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Note: The three periods in this definition are called ellipses and mean that you should
continue with the established pattern.
2) Empty set -The set containing no elements is called the empty set, or null set. The null set is
denoted by { }, or by the small Greek letter phi, ф.
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Note: The curly braces in this definition are called set braces.
There are 3 ways to define a set: (the definitions below all are defining the same set)
1. Word Description: Let A be the set of natural numbers less than 3.
2. Set-Builder Form:
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A = {x | x is a natural number, x<3}
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In this notation the curly braces are set braces.
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x is a variable, meaning it may take on a variety of values.
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The vertical bar stands for the phrase “with the property that.”
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A comma in this context means “and.”
3. Roster, or List Form:
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A = { 1, 2}
Section 1.1 - An Introduction to Sets
Let’s try these problems:

Write set C in roster form.
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
Write set B in roster form.
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
B = { k | 2(k+1)=6 } (this set is given in set builder form, which you may be familiar with)
Write the set D in set-builder form. (there are many correct answers, I will show 1 or 2 in class.)
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
C = { m | m is a natural number, m < 5}
D = {2, 3, 4, …}
Write a word description for the set F where: {𝑥| x is a natural number and 3 < 𝑥 ≤ 7}
A set should be well-defined.
A set is well-defined if any informed objective person can decide if a given element is in the set or
not.
Which of the following sets are well-defined?
A is the set of goofy dogs.
B is the set of GCC students whose gpa is 3.0 or greater.
C is the set of good GCC students.
D is the set of numbers whose square is 16.
Section 1.1 - An Introduction to Sets
Two sets are considered equal provided they have the same elements. The order the elements are
listed in is not important.
These two sets are equal: {1,2,3,4} = {4,1,2,3}
While these two sets are not equal: {1,2,3,4} ≠ {1,2,3}
More Notation – Consider the set A = {1,2,3,4,5}
The elements of the set A are 1,2,3,4 and 5. We write 5 ∈ A, which means 5 is an element of the set A.
If we write 6 ∉ A which means 6 is NOT an element of the set A.
Problems to complete in class: True or False
{1, 2} = {2,1}
ф={}
ф = {0}
{0, 1} = {1}
1 ∈ {1, 2}
0 ∈ {1, 2}
1 ∉ {1, 2}
0 ∉ {1, 2}
{1} ∈ {1, 2, 3, {1}}
{1} ∈ {1,2,3}
{1} ∉ {1, 2}
Section 1.2 - An Introduction to Sets part 2
Universal Set

A universal set for a particular problem is a set which contains all the elements of all the sets in
the problem.

A universal set is often denoted by a capital U.
Decide on an appropriate universal set U for the following problem situation. (there are many
possible answers)
A = {1, 2, 3}
B = {2, 4, 6, …}
C = {28}
Decide on an appropriate universal set U for the following problem situation.
A: The set of people who are currently enrolled in a math class at GCC.
B: The set of people enrolled in a physical education class at GCC.
C: The set of people enrolled in the nursing program at GCC.
Section 1.2 - An Introduction to Sets part 2
Finite Sets versus Infinite Sets

A set is finite if it is possible, given enough time, to write down every element in the set.

A set is infinite if it is not finite.
The Cardinal Number of a finite set A is the number of elements of A.
The cardinal number of a set A is denoted n(A).
Practice problems:
Find the cardinal number of the following set A.
(This question could have been written simply as Find n(A) = )
A = {x | x ∈ N, 2 < x < 10}
Find the number of elements (cardinal number) of set B.
B = The set of natural numbers less than 4.
Find n(C) where C = {a,e,I,o,u}
Set Equivalence for Finite Sets

Finite set A is equivalent to set B if n(A) = n(B). (that is the sets have the same number of
elements)

If two sets are equivalent it means that they can be put into one-to-one correspondence. Take
A={1,2,3} and B={a, b, c}. These two sets are equivalent because they both contain 3 elements.
1. Are sets A and B equal, equivalent or neither?
A = {apples, oranges}
B = {lemons, grapes}
2. Are sets C and D equal, equivalent or neither?
C = set of vowels in the English alphabet: (excluding y)
3. Are sets E and F equal, equivalent or neither?
4. Which of these are true? n(A) = n(B)
E = {1,3,5}
n(E) = n(F)
D = {a, e, i, o, u}
F = {2}
Section 1.2 - An Introduction to Sets part 2
Here are a few practice problems to check your understanding of the material covered to this point.
1. List the following sets in roster form.
a. A={x | x is a natural number, 2x=12}
b. B={k | k is a natural number, 3<k<8}
2. Is the set of scary cats a well-defined set? Why or why not?
3. True or False. n({x | x is a natural number less than or equal to 5})=n({w, x, y, z}).
Give a reason for your answer. (this question might feel like it is asked strangely, but it is a
common way to ask such a question)
4. Give an example of two sets which are equivalent but not equal. (many correct answers)
5. Why isn’t ф equal to the number 0?
Section 1.2 - An Introduction to Sets part 2
Definition of Subset
Set A is a subset of set B, symbolized A ⊆ B, if and only if all the elements of set A are also elements of
set B.
Note that A is a subset of set B if the following two conditions hold:
1. A is first and foremost a SET. (A can’t be a subset if it isn’t a set.)
2. If x ∈ A, then x ∈ B.
For the sets A and B below, is A a subset of B (that is A ⊆ B)
A = {a, e, i, o, u} B = {letters in the English alphabet}
Check the conditions:
1) Is A a set?
2) Are the letters a, e, i, o, and u contained in set B?
For the sets A and B below is B a subset of A (that is B⊆ A)? is A⊆B?
A = {1, 2, 3}
B = {2}
Fact: The empty set is a subset of every set.
Why? The reasoning is kind of hard to follow because you have to look at why it is that ф cannot not be
a subset of every set.
Suppose that there is some set A of which ф is not a subset. Then that means that there is something in
ф which is not in A. Since this can’t happen no such set A exists.
Section 1.2 - An Introduction to Sets part 2
Answer these True or False questions:

{1,2,3} = {3,2,1}
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{1,2,3} ⊆ {3,2,1}

1∈ {1,2,3}

1 ⊆ {1,2,3}
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{1} ⊆ {1,2,3}

ф ∈ {1,2,3}

ф ∈ { ф,{1,2,3},Fred}
Definition of Proper Subset
Set A is a proper subset of set B, symbolized by A ⊂ B, if and only if the following three conditions hold:
1. A is a set.
2. Every element of A is also an element of B.
3. A ≠ B.
Note: The first two conditions imply that A must be a subset of B. Therefore A is a proper subset of B if
A is a subset of B and A is not equal to B.
Answer the following True or False questions.
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{1,2,3} ⊂ {1,2,3}

{1,2} ⊂ {1,2,3}

φ ⊂ {1,2,3}

a ⊂ {a,b,c}

a ∈ {a,b,c}

{a} ⊂ {a,b,c}

{1} ⊄ {1}

φ=φ

{0} ⊄ { }
Section 1.2 - An Introduction to Sets part 2
Number of subsets of a set - Every set with n elements has 2n subsets.
Practice problems:
Let set A = {1,2}
How many subsets will A have?
List all the subsets of the set A = {1,2} (remember the empty set is a subset of every set)
Let set B = {1,2,3}
How many subsets will B have?
List all the subsets of the set B = {1,2,3}
How many subsets will the set E have if E = {1,2,3,4}
Section 1.3 Union and intersection of sets
Example. Let U be the set of natural numbers less than or equal to 10.
U = {1,2,3,4,5,6,7,8,9,10}
Let A={2,4,6} and B={1,2,3,4,5}.
Definition of Set Complement
The complement of set A, symbolized by A′, is the set of all the elements in the universal set that are not
in the set A.
A′ is read “A complement,” or “A prime.”
What is A’ for the sets A and U as described above?
What is B’ for the set B and U as described above?
Let U be the set of natural numbers less than or equal to 10.
Let C={2,4,7,8} and D={1,5}.
Find C′ and D′.
Definition of Set Intersection
The intersection of sets A and B, symbolized by A∩B, is the set containing all the elements that are
common to both A and B.
A∩B is read “A intersect B,” or “A AND B.”
Example - Let U be the set of natural numbers less than or equal to 10.
Let A={2,4,6} and B={1,2,3,4,5}.
Find A∩B.
Section 1.3 Union and intersection of sets
Example
Let U = {a, b, c, d}
A = {a, c}
B = {b,c, d}
1. Find A′.
2. Find B′.
3. Find A′ ∩ B.
4. Find A∩B.
5. Find (A∩B)′.
Definition of Set Union
The union of set A and set B, symbolized by A∪B, is the set containing all the elements that are
members of set A or of set B (or of both sets).
A∪B is read “A union B,” or “A OR B.”
Example. Let U be the set of natural numbers less than or equal to 10.
Let A={2,4,6} and B={1,2,3,4,5}.
Find A∪B.
Example
Let U = {a, b, c, d}
A = {a, c}
B = {b, d}
1. Find A∪B.
2. Find (A∪B )′.
3. Find A′ ∪ B.
4. Find (A′∩B)′.
Section 1.3 union and intersection of sets
Union Rule for Sets
The relationship between sets A, B, A∪B, and A∩B is given by the union rule:
n(A∪B) = n(A) + n(B) – n(A∩B)
Two Possible Union Rule Test Problems:
1. If n(A) = 5, n(B) = 8, and n(A∩B) = 2, find n(A∪B).
2. If n(A)=12, n(A∪B)=22, and n(A∩B)=10, find n(B).
Difference of Two Sets
The difference of two sets A and B, symbolized A-B, is the set of elements that belong to set A but not to
set B.
Example. Let U ={1,2,3,…,10}, A={2,4,6} and B={1,2,3,4,5}.
Find A-B and B-A.
A-B=
B-A=
Section 1.4 Venn Diagrams
A Venn diagram is a way to represent sets in picture form.
Construct a Venn diagram for the following sets.
U = {1,2,3,4,5,6,7,8,9}
A = {1,3,5,7,9}
B = {2,3,5,7}
C = {2,4,6,7}
Try to use your Venn diagram to construct the following sets:
𝐴 ∪ 𝐵′
(𝐴 ∪ 𝐶)′
(𝐴 ∩ 𝐶) ∪ (𝐵 ∩ 𝐶)
𝐴 ∩ (𝐵′ ∪ 𝐶)
Section 1.4 - Venn Diagrams
Example : A gym is thinking about adding more cardio equipment for it’s members. It is considering two types of
equipment, treadmills (T) and Stair Masters (S). The gym surveyed a sample of the members and asked which
equipment they used in the previous month. Of 150 gym members surveyed it was found that… 102 used the
treadmills 71 used the Stair Masters and 40 used both types.
Construct a Venn Diagram to model the problem.
Use your Venn diagram to answer the following.
Of those surveyed,
A) How many did not use either the treadmill or the Stair Master?
B) How many used the treadmill but not the Stair Master?
C) How many used the Stair Master but not the treadmill?
D) How many used either the treadmill or the Stair Master?
Section 1.4 - Venn Diagrams
Example 2: AAA Travel surveyed 125 potential customers. The following information was gathered.
68 wished to travel to Hawaii
53 wished to travel to Las Vegas
47 wished to travel to Disney World
34 wished to travel to Hawaii and Las Vegas
26 wished to travel to Las Vegas and Disney World
23 wished to travel to Hawaii and Disney World
18 wished to travel to all three destinations.
Create a Venn Diagram to answer the following questions. How many of those surveyed…
A) did not wish to travel to any of these destinations?
B) wished to travel only to Hawaii?
C) wished to travel to Disney World and Las Vegas, but not to Hawaii?
D) wished to travel to Disney World or Las Vegas, but not to Hawaii?
E) wished to only travel to exactly one of these locations?
Section 1.4 - Venn Diagrams
Which of the following are true statements for any sets A, B and C? Try to use
Venn Diagrams to justify your answers.
(𝐴 ∪ 𝐵)′ ∩ 𝐶 = 𝐴′ ∩ 𝐵′ ∩ 𝐶
𝐴 ∩ (𝐵 ∪ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
(𝐴 ∪ 𝐵 ∪ 𝐶)′ = 𝐴′ ∩ 𝐵′ ∩ 𝐶′