: Topic : Sets Objectives : - To understand the basic concept of sets - To understand the elements and members of a set - To understand the operations on sets such as union, intersection, complement and difference - To understand the basic sets such as natural numbers, integers, rational numbers and real numbers - To understand the concept of Venn diagrams Materials Required : Whiteboard, marker, textbook Procedure : 1. Introduction to sets (10 mins) - Explain what is a set and give examples - Explain the elements/members of a set - Explain the notations used for sets

1. FACULTY OF EDUCATION AND LANGUAGES
SEMESTER MAY / 2011
HBMT 4203
TEACHING MATHEMATICS IN FORM FOUR
MATRICULATION NO : 770218015450002
IDENTITY CARD NO. : 770218-01-5450
TELEPHONE NO. : 013-7018071
E-MAIL : znas77@yahoo.com.my
LEARNING CENTRE : JOHOR BAHRU

2. INTRODUCTIONS
SETS
What is sets in mathematics? A set is a collection of distinct objects, considered as an
object in its own right. Sets are one of the most fundamental concepts in mathematics.
Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics,
and can be used as a foundation from which nearly all of mathematics can be derived. In
mathematics education, elementary topics such as Venn diagrams are taught at a young age,
while more advanced concepts are taught as part of a university degree.
SETS THEORY
Set theory is the branch of mathematics that studies sets, which are collections of
objects. Although any type of object can be collected into a set, set theory is applied most
often to objects that are relevant to mathematics. The language of set theory can be used in
the definitions of nearly all mathematical objects.
The modern study of set theory was initiated by Georg Cantor and Richard Dedekind
in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems
were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with
the axiom of choice, are the best-known.
Concepts of set theory are integrated throughout the mathematics curriculum in the
United States. Elementary facts about sets and set membership are often taught in primary
school, along with Venn diagrams, Euler diagrams, and elementary operations such as set
union and intersection. Slightly more advanced concepts such as cardinality are a standard
part of the undergraduate mathematics curriculum.
Set theory is commonly employed as a foundational system for mathematics,
particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its
foundational role, set theory is a branch of mathematics in its own right, with an active
research community. Contemporary research into set theory includes a diverse collection of
topics, ranging from the structure of the real number line to the study of the consistency of
large cardinals.

3. SETS HISTORY
Mathematical topics typically emerge and evolve through interactions among many
researchers. Set theory, however, was founded by a single paper in 1874 by Georg Cantor:
"On a Characteristic Property of All Real Algebraic Numbers".[1][2]
Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the
West and early Indian mathematicians in the East, mathematicians had struggled with the
concept of infinity. Especially notable is the work of Bernard Bolzano in the first half of the
19th century. The modern understanding of infinity began in 1867-71, with Cantor's work on
number theory. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's
thinking and culminated in Cantor's 1874 paper.
Cantor's work initially polarized the mathematicians of his day. While Karl
Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of
mathematical constructivism, did not. Cantorian set theory eventually became widespread,
due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his
proof that there are more real numbers than integers, and the "infinity of infinities" ("Cantor's
paradise") the power set operation gives rise to.
The next wave of excitement in set theory came around 1900, when it was discovered
that Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes.
Bertrand Russell and Ernst Zermelo independently found the simplest and best known
paradox, now called Russell's paradox and involving "the set of all sets that are not members
of themselves." This leads to a contradiction, since it must be a member of itself and not a
member of itself. In 1899 Cantor had himself posed the question: "what is the cardinal
number of the set of all sets?" and obtained a related paradox.
The momentum of set theory was such that debate on the paradoxes did not lead to its
abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the
canonical axiomatic set theory ZFC, which is thought to be free of paradoxes. The work of
analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory.
Axiomatic set theory has become woven into the very fabric of mathematics as we know it
today.

4. SETS DEFINITION
Georg Cantor, the founder of set theory, gave the following definition of a set at the
beginning of his Beiträge zur Begründung der transfiniten Mengenlehre.
A set is a gathering together into a whole of definite, distinct objects of our perception and of
our thought - which are called elements of the set.
The study of algebra and mathematics begins with understanding sets. A set is
something that contains objects. To be contained in a set, an object may be anything that you
want to consider. An object in a set may even be another set that contains its own objects. In
everyday language, a set can also be called a “collection” or “container”, but in mathematics,
the term set is preferred. The objects that a set contains are called its members.
The elements or members of a set can be anything: numbers, people, letters of the
alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. Sets A
and B are equal if and only if they have precisely the same elements.
As discussed below, the definition given above turned out to be inadequate for formal
mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set
theory, and its properties are defined by the Zermelo–Fraenkel axioms. The most basic
properties are that a set "has" elements, and that two sets are equal (one and the same) if and
only if they have the same elements.
CONCEPTS
A set A consists of distinct elements :
If such elements are characterized via a property E, this is symbolized as follows:
satisfies property E}.

5. The following notations are commonly used:
notation meaning
is element / member of
is not element / member of
is a subset of
is a strict subset of
number of elements in
empty set
If ( ), is called a finite (infinite) set.
Two sets are called equipotent, if there exists a bijective map between their elements (
for finite sets and ).
The set of all subsets of is called power set, i.e. . . In
particular, we have and . Moreover, .
The following sets are standardly denoted by the respective symbols:
natural numbers:
integers:
rational numbers:
real numbers:
complex numbers:
The following notations are also commonly used and as
as well as , ,,, ,, respectively.

6. OPERATIONS ON SETS
The following operations can be applied to sets and :
Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of
A, or B, or both. The union of {1, 2, 3} and {2, 3, 4} is the set {1, 2, 3, 4}.
Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are
members of both A and B. The intersection of {1, 2, 3} and {2, 3, 4} is the set {2, 3} .
Set difference, complement of U and A, denoted U A is the set of all members of U
that are not members of A. The set difference {1,2,3} {2,3,4} is {1} , while,
conversely, the set difference {2,3,4} {1,2,3} is {4} . When A is a subset of U, the
set difference U A is also called the complement of A in U. In this case, if the choice
of U is clear from the context, the notation Ac is sometimes used instead of U A,
particularly if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B is the set of all objects that are a member of
exactly one of A and B (elements which are in one of the sets, but not in both). For
instance, for the sets {1,2,3} and {2,3,4} , the symmetric difference set is {1,4} . It is
the set difference of the union and the intersection, (A ∪ B) (A ∩ B).
Cartesian product of A and B, denoted A × B, is the set whose members are all
possible ordered pairs (a,b) where a is a member of A and b is a member of B.
Power set of a set A is the set whose members are all possible subsets of A. For
example, the power set of {1, 2} is { {}, {1}, {2}, {1,2} } .
Some basic sets of central importance are the empty set (the unique set containing no
elements), the set of natural numbers, and the set of real numbers.

7. The so- called Venn diagrams illustrate the set operations.
union:
Intersection: difference: symmetric difference:
If , some of the above diagrams are identical to one another:
Union: intersection: complement set:

8. LESSON PLAN:
Day : Thursday
Date : 30 June 2011
Class : 4 Bukhari
Subject : Mathematics :
Time : 10.00 am – 11.20 am
Duration : 80 minutes
Learning Area : 3) Sets
Learning objectives : Students will be taught to :
3.1 Understand the concept of sets
Learning outcomes : Students will be able to :
(i) represents sets by using Venn diagrams
Teaching aids : Manila cards ((Closed Geometrical shaped), activity sheets,
mahjong papers, quiz papers, LCD, worksheets.
Attitudes and Values : Patient, self-confident, concentrate, cooperative, follow instructions,
honesty, careful
Thinking Skills : Categorize, recognize the main idea, making sequence to represent
sets by using Venn diagram, locate and collect relevant information,
analyze part / whole relationships, reflection.
Previous Knowledge : i) sort given objects into groups
ii) define sets by description and using set notation
iii) identify whether a given object is an element of a set

9. TEACHING AND LEARNING
STEPS CONTENTS NOTES
ACTIVITIES
Step 1 Parts of “Definition of Set” Teacher shows an example Teaching
Revision Categorize set A on whiteboard. Aids:
Manila Cards
Example: Students pay attention on the
A = {Factors of 30) example written on the whiteboard. Values:
A = {1,2,3,5,6,10,15,30} Self
Teacher wants the students to look at confident,
the Set A and try to list out all the honest,
elements by using set notation. patient.
Students try to list out all the Thinking
elements of Set a by using set Skills:
notation. Categorize
Teacher calls a student to give an
answer on the whiteboard.
A student tries to give an answer.
Teacher checks the answer.

10. STEPS CONTENTS TEACHING AND LEARNING NOTES
ACTIVITIES
Method:
Step 2 Venn Diagram Teacher introduces Venn diagram to
Explanation
Besides the methods of the students and explains that it is
description and set notation, easier to see which group each
Teaching
sets can be representing by element belongs to in a Venn
Aids:
using Venn Diagrams. diagram.
Manila cards
(Closed
Closed Geometrical Shapes Students pay attention on
Geometrical
explanation and the given examples
Shapes)
of Closed Geometrical Shapes
Circle Oval Rectangle
Values:
Teacher stress that rectangle are
Concentrate
usually used to represent the set
which contain all the elements that
Square Triangle Hexagon Thinking
are discussed and the circles or
Skills:
enclosed curves to represent each set
Recognize
within it.
the main
idea,
Teacher gives examples of Closed
making
Geometrical Shaped.
sequence to
represent sets
by using
Venn
diagram

11. STEPS CONTENTS TEACHING AND LEARNING NOTES
ACTIVITIES
Step 2 Examples: Then, teacher uses the example from Vocabulary:
(continued) induction set and teaches the students Set
(1)
A={1, 2, 3, 5, 6, 10, 15, 30} the steps to draw a Venn diagram to Element
Each ‘dot’
A •1 •2 represents
represent set A as listed below: Description
•3 •5 •6 one element 1. Draw a circle. Label
•10 •15
•30 2. Represent set A by Set notation
labeling the circle as A. Denote
(2) 3. Determine the number of Venn
B={a, b, c} elements in set A, and Diagram
B •a represent each of them Empty set
•b
with a dot inside the circle. Equal sets
•c
Subset
Students concentrate on showing set Universal set
(3)
Q={Multiples of 3 between in Venn diagram. Complement
8 and 18}
of a set
Q={9, 12, 15}
Teacher reminds students to put a Intersection
Q •9
•12 “dot” to present one element and Common
•15 label the set. elements
Students remember the important
point.
Teacher shows another two examples
- (2) and (3).
Teacher asks a student to put
elements into diagram.

12. STEPS CONTENTS TEACHING AND LEARNING NOTES
ACTIVITIES
Student tries to put elements into
diagrams.
For example no. (3), teacher asks a
student to list out elements first then
put the elements into Venn diagram.
Another student has to do example
no. (3).
Teacher gives time to copy notes.
Students copy the notes.
Step 3 Teaching Progression After copy the given notes, teacher Values:
enhance students understanding with Cooperative,
Teacher conduct the group conduct group activity calls “fast and Follow
activity to further enhance correct”. Instructions
student‟s understanding about
the lesson learnt today. Teacher explains the rules of the Teaching
group activity: Aids:
Group Activity “Group that can answer all from 4 Activity
1. Given that W is a set questions fast and correct, will be the sheets,
representing days of a week, winner”. Mahjong
draw a Venn diagram Paper
representing the elements of Students follow the rules in the
W. activity.

13. STEPS CONTENTS TEACHING AND LEARNING NOTES
ACTIVITIES
2. Teacher observes students to do the Thinking
If, activity. Skills:
s x : x is an odd numbered Locate and
20 x 3o , Can you Students ask questions if not collect
understand. relevant
draw a Venn diagram to
information.
represent the elements in
Teacher wants one representative
this set?
from each group to come up
randomly to check the solution from
3. Draw a Venn diagram to
other group on the mahjong paper.
represent the set given
Example: every first member of the
below.
group will check number 1, second
member from each group to do
P x : x is a prime number ?
number 2 and so on.
i) 1 x 10
ii) 41 x 50
Each group writes down their
solution on the mahjong paper,
4. Given that B is the set of
depends on the problem solving.
common factors of 24 and
36.
Finally, teacher discusses the
Draw a Venn diagram of set
solution with the students.
B.
Students do the corrections if they
make any mistakes.

14. STEPS CONTENTS TEACHING AND LEARNING NOTES
ACTIVITIES
Step 4 Quiz Teacher distributes each student a Teaching
(1) Given that P = {1, 2, 3, 4, quiz paper (with two questions). Aids:
5}. Represent set P by Quiz Paper,
using Venn diagram. Students get a piece of quiz paper. LCD
Solution: Teacher asks students try to draw a Values:
diagram without asking friends or Honest,
P •1
teacher. Careful
•2 •3 •4
•5
Students try to draw a Venn diagram Thinking
by themselves. Skills:
(2) If R = {x : 30 ≤ x ≤ 40, x is
Analyze
a multiple of 3}, can you
(Teacher do not forget gives guide relationships.
draw a Venn diagram to
line to the students)
represent the elements in
this set?
Teacher observes students to do
quiz.
Solution:
R = {30, 33, 36, 39}
Teacher collects papers after three
minutes.
R •30
•33 •39
Students pass up quiz papers.
•36
Teacher discusses the answers for
the quiz with the students and guides
them.
Students respond to the teacher and
listen to the answer.

15. STEPS CONTENTS TEACHING AND LEARNING NOTES
ACTIVITIES
Step 5 Summary and exercises Teacher asks students to make a Teaching
on Venn diagram. summary of the day‟s lesson. Aids:
Conclusion Worksheets
Students make summary of the day‟s
Venn Diagram
lesson. Values:
Self
Teacher reminds students what they Confident
Representing in have learnt how to draw a Venn
Closed Geometrical
Shaped diagram to represent the elements of Thinking
a set. Skills:
Reflections
All the elements Teacher will stress that to put a
in a SET “dot” to present one element.
Students remember the important
Each element will have points.
a DOT beside it (left)
Teacher distributes the activity
sheets.
Students do the worksheets and
exercises given.

16. TEACHING AIDS

21. WORKSHEETS 1
NAME:_____________________________________ DATE: ______________
FORM: 4 _________________
Answer all questions.
1. Given that Z = {multiple of 3}, determine if the following are elements pf set Z. Fill in the
following boxes using the symbol or .
a) 52 Z b) 18 Z c) 69 Z
2. Y is a set of the months that start with the letter “M”. Define set Y using set notation.
3. Determine whether 8 is an element of each of the following set.
a) {2,4,6,8}
b) {Multiples of 4}
c) {LCM of 2 and 4)

22. d) {HCF of 4 and 8}
4. State whether each of the following sets is true or false.
a) 4 {Common factors of 8 and 12}
b) 1 {Prime number}
5. State the number of elements of each of the following.
a) X = {Cambodia, Singapore, Malaysia, Indonesia, Thailand}
b) Y = {3,6,9, …21)
c) Z = {Integers between -3 and 4, both are inclusive}

23. WORKSHEETS 2
NAME:_____________________________________ DATE: ______________
FORM: 4 _________________
Answer all questions.
1. Draw a Venn diagram to represent each of the following:
a) F = {1,3,5,7}
b) M = {Pictures of durian, mangosteen, mango, starfruit}
c) N = {The first 5 prime numbers}
2. Use the notation {} to represent set A, B and C.
a) A
1
4
9

24. b) B
h
j
k
l
m
n
p
r
c) C
100
250
150
200
3. State n(A) when
a) A = {The letters in the word „SCIENCE‟}
b) A = {x : 5 x < 30 where x is not an odd number}
c) A is a set of perfect square numbers between 0 to 90.

25. CONCLUSION
This assignment shows us the introduction that explain on what is sets, the sets theory