This document defines sets and their properties. It begins by defining a set as a well-defined collection of objects and notes that sets are conventionally denoted with capital letters while elements are denoted with lowercase letters. It then discusses different ways of representing sets, including statement form, roster form, and set builder form. Venn diagrams are introduced as a pictorial way to represent sets and relationships between sets. Various set symbols and their meanings are defined. Different types of sets such as singleton, finite, and infinite sets are described. Properties of sets like commutative, associative, distributive, identity, complement, idempotent, bound, and absorption laws are explained. Finally, operations on sets like union, intersection, difference,

1. SETS & THEIR
PROPERTIES
BY : GROUP 2

2. DEFINITION
A set is a well defined collection of objects. The objects that make
up a set (also known as the elements or members of a set) can be
anything : numbers, people, letters of the alphabet, other sets,
and so on. Georg Cantor, the founder of set theory.
Sets are conventionally denoted with capital letters,and elements
of a set are usually denoted by lower case letters. The notation,
𝒂 ∈ 𝑨, means that a is a member of the set A. A set can be defined
by listing its elements inside braces. For example :
1. Let set A be {all animals with four feet}
2. Let set B be {positive integers, more than 2}
3. Let set C be {1, 3, 5, 7,….}
4. Let set D be {the first five odd numbers}

3. PRESENTATION OF SETS
There are different set notations used for representation of
sets. They differ in the way in which the elements are listed. The
three set notations used for representing sets are :
1. Statement Form
The statement form, describes a statement to show what are the
elements of a set. Properties of a member of a set are written
and enclosed within curly brackets. For instance,
a. The set of even numbers is less than 20. In the statement from
method, it is represented as {even numbers less than 20}.
b. Set A is the list of the first five odd numbers. It is
represented as A = {first five of odd numbers}.

4. PRESENTATION OF SETS
There are different set notations used for representation of
sets. They differ in the way in which the elements are listed. The
three set notations used for representing sets are :
2. Roster Form
The most common form used to represent sets is the roster
notation in which the elements of the sets are enclosed in curly
brackets separated by commas. To sum up the notation of the
roster form, please take a look at the examples below.
a. Finite Roster Notation of Sets : Set A = {1, 2, 3, 4, 5} (The first
five natural numbers) .
b. Infinite Roster Notation of Sets : Set B = {5, 10, 15, 20 ....} (The
multiples of 5).

5. PRESENTATION OF SETS
There are different set notations used for representation of
sets. They differ in the way in which the elements are listed. The
three set notations used for representing sets are :
3. Set Builder Form
The set builder notation begins with an alphabet, say x, as a
variable, followed by a colon. Then all the properties that an
element x must satisfy to be considered a member of the set are
then written. The set builder form uses a vertical bar in its
representation, with a text describing the character of the
elements of the set. For example, A = { k | k is an even number, k ≤
20}. The statement says, all the elements of set A are even
numbers that are less than or equal to 20. Sometimes a ":" is used
in the place of the "|".

6. VENN DIAGRAM
Venn Diagram is a pictorial representation of sets, with
each set represented as a circle. The elements of a set
are present inside the circles. Sometimes a rectangle
encloses the circles, which represents the universal
set. The Venn diagram represents how the given sets
are related to each other. Venn Diagrams or set
notation can be used to show the relationship between
sets.

7. VENN DIAGRAM
RELATIONS VENN DIAGRAM SET NOTATION
Set B is a proper subset of set A. Set B in
contained in set A.
𝐵 ⊂ 𝐴
Set 𝐵𝑐
is a subset of set U 𝐵𝐶
⊂ 𝑈
Set I added to set K is the union of I and K 𝐼 ∪ 𝐾

8. VENN DIAGRAM
Set I subtracted from set K, is the difference
between I and K.
𝐾 − 𝐼
The members common to set I and K from
the intersection of I and K.
𝐾 ∩ 𝐼
Sets B and D are disjoint 𝐵 ∩ 𝐷 = ∅ 𝑜𝑟 {}

9. SETS SYMBOLS
Set symbols are used to define the elements of a given set. The following
table shows some of these symbols and their meaning.
SYMBOLS MEANING
∪ Universal Set
𝑛(𝑋) Cardinal number of set X
𝑏 ∈ 𝐴 ‘b’ is an element of set B
𝑎 ∉ B ‘a’ is not an element of set B
{ } Denotes a set
∅ Null or empty set
𝐴 ∪ 𝐵 Set A union set B
𝐴 ∩ 𝐵 Set A intersection set B
𝐴 ⊆ B Set A is a subset of set B
𝐵 ⊇ A Set B is the superset of set A

10. TYPES OF SETS
1
SINGLETON SETS
2
FINITE SETS
3
INFINITE SETS
4
EMPTY OR NULL SETS
5
EQUAL SETS
6
UNEQUAL SETS

11. TYPES OF SETS
7
EQUIVALENT SETS
8
OVERLAPPING SETS
9
DISJOINT SETS
10
SUBSET AND
SUPERSET
11
UNIVERSAL SETS
12
POWER SETS

12. PROPERTIES
OF SETS
There are nine important
properties of sets. Given, three
sets A, B, and C, the properties
for these sets are as follows.
Where set A, B, and C are not
empty set or ∅ 𝑜𝑟 .
PROPERTY EXAMPLE
Commutative Law
𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴
𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴
Associative Law
𝐴 ∩ 𝐵 ∩ 𝐶 = 𝐴 ∩ 𝐵 ∩ 𝐶
(𝐴 ∪ 𝐵) ∪ 𝐶 = 𝐴 ∪ (𝐵 ∪ 𝐶)
Distributive Law
𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶
𝐴 ∩ 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶
Identity Law
𝐴 ∪ ∅ = 𝐴
𝐴 ∩ 𝑈 = 𝐴
Complement Law
𝐴 ∪ 𝐴 = 𝑈
𝐴 ∩ 𝐴 = ∅
Idempotent Law
𝐴 ∪ 𝐴 = 𝐴
𝐴 ∩ 𝐴 = 𝐴
Bound Law
𝐴 ∪ 𝑈 = 𝑈
𝐴 ∩ ∅ = ∅
Absorption Law
𝐴 ∪ 𝐴 ∩ 𝐵 = 𝐴
𝐴 ∩ 𝐴 ∪ 𝐵 = 𝐴
De Morgan Law
𝐴 ∪ 𝐵 = 𝐴 ∩ 𝐵
𝐴 ∩ 𝐵 = 𝐴 ∪ 𝐵

13. 1
2
3
4
5
OPERATION
ON SETS
Some
important operation
s on sets include
union, intersection,
difference, the
complement of a
set, and the
cartesian product
of a set. A brief
explanation of
operations on sets
is as follows.
UNION OF SETS
INTERSECTION OF
SETS
SETS DIFFERENCE
SETS COMPLIMENT
CARTESIAN PRODUCT
OF SETS

14. EXERCISE 1
Write the set of vowels used in the
word ‘UNIVERSITY’.

15. EXERCISE 2
For each statement, given below, state
whether it is true or false along with the
explanations.
i. {9, 9, 9, 9, 9, ……..} = {9}
ii. {p, q, r, s, t} = {t, s, r, q, p}

16. EXERCISE 3
Look at this set of the number below ;
2, 3, 5, 7, 11, 13, 17, 19, 23,….
a. How many elements of prime number?
b. Can you find one even number on this set
number?

17. EXERCISE 4
Pictures
Here
Suppose that in a town, 800 people are selected
by random types of sampling methods. 280 go to
work by car only, 220 go to work by bicycle only
and 140 use both ways – sometimes go with a car
and sometimes with a bicycle. Here are some
important questions we will find the answers:
a. How many people go to work by car only?
b. How many people go to work by bicycle only?
c. How many people go by neither car nor
bicycle?
d. How many people use at least one of both
transportation types?
e. How many people use only one of car or
bicycle?

18. Thank You
Any questions?