2. What is a set?
A set is a well-defined collection of objects called elements.
Below is an example of a set, a set of tools:
Tools
Hammer
Sickle
Pliers
Scissors
3. The List notation
The list notation is a method of identifying the elements of a set by
listing the individual elements.
In the previous example, we write the set of tools as:
Tools = {Hammer, Sickle, Pliers, Scissors}
Name of the set Elements
• The elements are separated by commas “,”
• The curly braces “{ }” specify the bounds of the set
4. Activity 1: Set it!
Directions: - Time limit : 5 minutes.
1. List down 5 examples of sets that can be easily seen in your house.
2. Encode your answers using MS Word.
3. Submit your answers to my email.
5. Discussion:
• What sort of sets did you discover?
• How did you find this activity? Was it easy or hard to identify sets?
6. Additional points to remember about sets:
1. Sets can either be finite or infinite.
e.g.: A={1,2,3,4} B={1,2,3,…}
*the three dots is called an ellipsis, meaning that the earlier pattern will be
continue infinitely.
Set B is also known as the set of Whole numbers.
7. Additional points to remember about sets:
(cont.)
2. Repeated elements are only listed once.
e.g.: {a,b,a,c,b,a} is the same with: {a,b,c}
8. Additional points to remember about sets:
(cont.)
3. There is no order in a set.
e.g. A = {1,2,3,4,5}
Set A can also be written as:
A = {5,4,3,2,1} = {2,3,4,1,5} , etc.
9. Common Sets
The common sets have distinct characters due to their properties.
These sets are widely used and referred to in mathematics:
• Natural Numbers: N={0,1,2,3,…}
• Integers: Z= {…,-2,-1,0,1,2,…}
• Rational Numbers: Q= {1/1, ½, 1/3, ¼, …}
• Irrational Numbers: I= {e, π, √3, √5,…}
• Real Numbers: R= {{N}, {Z}, {Q}, {I}}
• Complex Numbers: C= {i,2i, 3i,...}
10. Elements and cardinality
Using our example in the second slide,
Let: T={Hammer, Sickle, Pliers, Scissors} * ε is the Greek letter epsilon*
*Cardinality refers to the number of elements*
Hammer is an element of T ----------------------------- “Hammer ε T”
Paint is not an element of T ----------------------------- “Paint ∉ T”
The Cardinality of T is 4 ------------------------------ “|T| = 4”
11. The null set, or empty set
The empty set is a set that contains nothing, it is represented by the
character ∅ .
∅ = { }
Since the null set contains nothing, its cardinality is:
|∅|=0
12. Set-builder notation
a method of describing a set by stating the form and properties that its
elements must satisfy.
e.g. 1:
Let: A = A = {1,2,3,4,5}; in set builder notation - A = {x| x ε Z+, x<6}
The statement above translates to A contains all element x, such that x
is a positive integer, and x is less than 6.
13. Set-builder notation (cont.)
e.g. no.2: Let B = {2,4,6,8,…}
B = {n| n ε Z+ , n/2 ε Z+}, which means B contains all elements n such
that, n is a positive integer, and n divided by 2 is also a positive integer.
---------------------------------------------------------------------------------------------
B = {2n| n ε Z+}, which means B contains all elements 2n, such that n is
a positive integer.
*there are many ways to write the set-builder notation. For as long as it
correctly describes the properties of the set.
14. Set-builder notation (cont.)
Set builder notation can also be used on sets that are easily seen in our
surroundings!
e.g.: write a list notation of this set:
In set builder notation:
School Supplies = {x|x can be used at school}
Meaning, School supplies contain all elements x such that x can be
used at school
15. Exercises:
Let: D = {x ε Z+| x<6}
• List the Elements of set D using the list notation
• Determine the Cardinality of Set D
• Determine the cardinality of this set:
B= {a, b, {c,d}}
16. Answers:
Let: D = {x ε Z+| x<6}
1. List the Elements of set D using the list notation
Answer:
The set builder notation of set D translates to “D contains all elements
x that are positive integers such that x is less than 6. This means all
positive integers less than 6 are elements of set D.
D= {1,2,3,4,5}
17. Answers (cont.)
Let: D = {x ε Z+| x<6}
2. Determine the Cardinality of Set D
Answer:
The list notation of set D is: D={1,2,3,4,5}
By inspection, the cardinality is:
|D| = 5
18. Answers (cont.)
3. Determine the cardinality of this set:
B= {a, b, {c,d}} Answer: |B|=3.
The explanation for this answer can be deduced using this graphical
representation of Set B:
19. Practical applications of sets
The set theory has various applications not only in the field of
mathematics but also in other areas like the sciences and in computer
programming. The study of sets is necessary in preparation for a
student undertaking higher mathematics in the future.
20. Summary of today’s Lesson
1. A set is a well defined collection of objects called elements.
2. Examples of common sets are; the set of Real numbers R, the set of
Complex numbers C, the set of Rational numbers Q, the set of
Irrational numbers I, the set of Integers Z, and the set of Natural
numbers N.
21. Summary (cont.)
3. The elements of a set can be determined by inspection using the list
notation, or by analysis of the set-builder notation.
4. The cardinality refers to the amount of distinct elements in a set.
5. The set-builder notation is used to easily generalize the elements of
any set by specifying the element’s form and its properties.
23. Online Quiz - Send your answers to my email.
1. State the definition of set
2. Enumerate at least three common sets, and list down some
examples of its elements using the list notation.
3. Determine the elements of set C:
C = {x|x ε Z, 1 ≤ x ≤ 6}
4. Determine the cardinality of set C.
24. Online quiz
5. Rational numbers are numbers that can be written as fractions.
Use the set-builder notation to describe the elements that belong to
the set of Rational numbers Q.
