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Lesson 3 - Introduction to Sets

Learning Outcomes
By the end of this lesson, students will be able to;

• define a set for a given collection of objects.

• representation of sets using builder form and tabular form.

• define finite sets, infinite sets, null sets, subsets, proper set, equality of sets and disjoint
sets.

• compute the cardinality of a finite set.

3.1 Introduction
Set is defined as a collection of well-defined objects. The objects in a set are called the elements
or members, of the set. The elements in a set are of the same kind and they are distinct with
no repetition of the same element in the set. Usually capital letters, A, B, X, Y , . . . , are used
to denote sets and lowercase letters a, b, x, y, . . . , are used to denote elements of sets. The
symbol ∈ is used to express that an element is belongs to a set and its negation is represented
by 6∈. For instance, 10 ∈ {12, 24, 10, 53} and 5 6∈ {12, 24, 10, 53}.

There are two main ways of specifying a set:

1. Tabular form
We can specify a set by explicitly listing all its elements.
Eg: A = {red, yellow, green, blue}

2. Set-builder notation
We can specify a set by giving a property that all elements must satisfy.
Eg: B = {x : x is a letter in the English alphabet, x is a vowel}
This would be read “B is the set of x such that x is a letter in the English alphabet and
x is a vowel”. Note that a letter, usually x, is used to denote a typical member of the set;
and the colon is read as “such that” and the comma as “and”.

3.2 Universal Set and Empty Set

A universal set is all the elements, or members, of any group under consideration. For example,
in astronomy all the stars of the Milky Way galaxy and in human population studies the
universal set consists of all the people in the world. The symbol U is used to denote the
universal set. The set with no element is called the empty set or null set. It is denoted by ∅ or
{}. For example,
n o
S = x : x is a positive integer, x2 = 3 .

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3.3 Some Important Sets

Some sets of numbers that occur frequently in mathematics. These sets and their notations are
as follows:

• N = {1, 2, 3, . . . } = the set of natural numbers

• N0 = {0, 1, 2, 3, . . . } = the set of natural numbers with zero

• Z = {. . . , −3, −2, −1, 0, 1, 2, 3, . . . } = the set of integers, both positive and negative, with
zero

• Z+ = {1, 2, 3, . . . } = the set of positive integers

• Z− = {−1, −2, −3, . . . } = the set of negative integers

n o
• Q= x:x= n
m
, n, m ∈ Z = the set of rational numbers
The numbers contained in Q are exactly those of the form m
n
, where n and m are integers
and m 6= 0.
3 8 136 −1
Eg: (= 1.5) ∈ Q, (= 2) ∈ Q, (= 1.36) ∈ Q, (= −0.001) ∈ Q
2 4 100 1000
• R = {x : −∞ < x < ∞} = the set of real numbers
√
Eg: 1.5, −12.3, 99, 2, π

• I = the set of imaginary numbers

• C = {z : z = a + bi, −∞ < a < ∞, −∞ < b < ∞} = the set of complex numbers

3.4 Subsets
Let A be a set. If B is a set such that each element of B is also an element of the set A, then
B is said to be a subset of the set A, denoted B ⊆ A or A ⊇ B.
Two sets are equal if they both have the same elements or, equivalently, if each is contained
in the other. That is:

A = B if and only if A ⊆ B and B ⊆ A.

Example 3.1
Consider the sets:

A = {1, 3, 4, 7, 8, 9} , B = {1, 2, 3, 4, 5} , C = {1, 3}

Then C ⊆ A and C ⊆ B since 1 and 3, the elements of C, are also members of A and B.
But B 6⊆ A since some of the elements of B, e.g., 2 and 5, do not belong to A. Similarly,
A 6⊆ B.

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Remark 3.1.
1. For any set A, we have ∅ ⊆ A ⊆ U .
2. For any set A, we have A ⊆ A.
3. If A ⊆ B and B ⊆ C, then A ⊆ C.

Remark 3.2. Observe that

N ⊆ Z ⊆ Q ⊆ R ⊆ C.
This is shown in Figure 3.1.

Figure 3.1: Number sets

3.5 Proper Subset

If A ⊆ B, but A 6= B, we say A is a proper subset of B, denoted A ⊂ B.

Example 3.2

Let C = {1, 3} and A = {1, 2, 3, 4}. Then C is a proper subset of A since C is a subset
of A but C does not equal A. We write C ⊂ A.

3.6 Disjoint Sets

Two sets A and B are said to be disjoint if they have no elements in common.

Example 3.3
Suppose
A = {1, 2} , B = {4, 5, 6} and C = {5, 6, 7, 8} .
Then A and B are disjoint and A and C are disjoint. But B and C are not disjoint since
B and C have elements in common, e.g., 5 and 6. We note that if A and B are disjoint,
then neither is a subset of the other

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3.7 Finite and Infinite Sets

Sets can be finite or infinite. A set S is said to be finite if S is empty or if S contains exactly
m elements where m is a positive integer; otherwise S is infinite.
If the set S is finite, its number of elements is called as “cardinality” and it is represented as
|S| or n(S).

Example 3.4

(i) If A = {12, 24, 10, 53} then n(A) = 4. Therefore A is a finite set.

(ii) Let E be the set of even positive integers, that is E = {2, 4, 6, . . . }. Then E is
infinite.

3.8 Venn Diagram

A Venn diagram is a pictorial representation of sets in which sets are represented by enclosed
areas in the plane. The universal set U is represented by the interior of a rectangle and the other
sets are represented by disks lying within the rectangle. Figure 3.2 shows the Venn diagrams
for disjoint sets and proper subsets.

Figure 3.2: Venn diagrams

Self-Assessment Exercises
1. List the elements of each set where N = {1, 2, 3, . . . }.

(a) A = {x ∈ N : 3 < x < 9}

(b) B = {x ∈ N : x is even, x < 11}
(c) C = {x ∈ N : 4 + x = 3}

2. Decide whether the following statements are true or false.

(a) {2, 4, 7} ⊂ {3, 2, 5, 4, 7}

(b) {2, 3, 0} ⊆ N
(c) −5 ∈ N

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3. Let A = {2, 3, 4, 5}.

(a) Show that A is not a subset of B = {x ∈ N : x is even}.

(b) Show that A is a proper subset of C = {1, 2, 3, . . . , 8, 9}.

Suggested Reading
Chapter 2: Kenneth Rosen, (2011) Discrete Mathematics and Its Applications, 7th Edition,
McGraw-Hill Education.