ST. GABRIEL THE ARCHANGEL HIGH SCHOOL, INC.

San Gabriel, La Union

Tel., No. (072)-682-0082
“A Seed by Love for All, Dedication to theCommunity and Commitment to the Child’’
SCHOOL ID: 400121 SCHOOL ESC ID:0100071

Name: ______________________________________ QUARTER 1

Section: _______________________ MATHEMATICS 7
Module 1

Content Standards: The learner demonstrates understanding of key concepts of sets and the real
number system.
Performance Standards: The learner is able to formulate challenging situations involving sets and real
numbers and solve these in a variety of strategies.
Learning Competencies: The learner
i. Illustrates well – defined sets, subsets, universal sets, null set, cardinality of sets, union and
intersection of sets and the difference of two sets; and
ii. Solves problems involving sets with the use of Venn diagram.

UNIT I: SETS AND REAL NUMBERS

CHAPTER 1: THE REAL NUMBER SYSTEM

There are many benefits of travelling in groups – just ask the animals! Aside from cooperative food
collection, sharing of critical information and protection from predators, being in a group makes
travelling fun, too!
The images show a colony of penguins, cast of crabs, a pod of dolphins, and a tower of giraffes.
‘Colony’, ‘cast’, ‘pod’, and ‘tower’ are collective nouns for group of animals. What names for other group of
animals do you know?
The notion of a set is not exactly new. All of us have many experiences with groups of things. This
chapter discusses the language of sets and set operations.

Lesson 1.1 INTRODUCTION TO SET THEORY

Prerequisite
Determine which element (or member) does not belong to the given collection.
1. { goat , carabao , deer , dog , cow }
2. { 6 , 8 , 9 , 10 ,12 }
3. { January , March , Wednesday , August , December }
4. { yellow , blue , apple , red , green }
5. { adobo ,menudo , kare−kare , leche flan , sinigang }

The concept of a set is important in the study of Mathematics and other related fields.
WELL-DEFINED COLLECTIONS AND SETS
A collection is well-defined if, given an object, it can be categorically identified to belong to the
collection or not.
Consider these collections of objects.
Column A Column B
Collection of positive integers Collection of rich people
Collection of Math books in the library Collection of large animals
Collection of names of students in a school Collection of nice objects
Collection of animals weighing at least 300 kg Collection of bright students

In Column A above, each collection is well-defined because one can identify if an object belongs to
the collection or not. In Column B, phrases like “rich people”, “large animals”, “nice objects”, and “bright
students” are not well-defined because these are subjective phrases.

EXAMPLE 1.1.1

A group or collection of objects is called a set. Each object in a set is called a member or an
element of a set.

SET NOTATION
There are three ways to describe sets:
1. The descriptive method
A set can be described by writing a description of its elements.
For example:
V = colors in the Philippine flag
M = multiples of 6 between 1 and 400

2. The list or roster method

A set can be described by listing all its elements within a pair of braces, { }. Each element is
separated from the next by a comma.

Set V can be written as

V = { blue ,red , white , yellow }

Set M can be written as

M = {6 ,12 , 18 , … ,396 }.

3. The set-builder notation

A set can be described using variables. A variable is a symbol, usually a letter that can represent
different elements of a set. For example, suppose you are interested in all the cities of the
Philippines. Since there are too many elements in this set, it would be better to use the set-builder
notation.
A={ x|x is a city ∈the Philippines }
Read as “A is the set of all x such that x is a city in the Philippines”.

EXAMPLE 1 Write each set using the roster method.

a. The set of distinct letters in the word HUMILITY
b. The set of colors of the rainbow
SOLUTIONS: a. { H , U , M , I , L ,T ,Y }
Notice that when listing the elements of a set, identical elements are not repeated.
b. { red , orange , yellow , green ,blue , indigo , violet }

EXAMPLE 2 Write each set using the descriptive method.

a. { 1 , 2, 3 , 4 ,5 , 6 , 7 , 8 , 9 ,10 }
b. { September , October , November , December }
SOLUTIONS: a. { first 10 counting numbers }
b. the set of months ending in ‘ber’

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EXAMPLE 3 Write each set using set builder notation.
a. { 1 , 4 , 9 , 16 , 25 }
b. { red , yellow , blue }
SOLUTIONS: a. { x|x is a perfect square }
b. ¿

EXAMPLE 4 Write the set10, 11, 12, 13, … using

a. the roster method
b. the descriptive method
c. set-builder notation
SOLUTIONS: a. { 10 , 11, 12, 13 , … }
b. the set of counting numbers greater than 9
c. { { x |x is a cunting number∧x > 9 }

1.1.2 ELEMENTS OF A SET AND WELL DEFINED SET

ACTIVITY NO. 1
Are you a member of the set?
Stand up if you are a member of this set.
1. Set of students who were born in June
2. Set of students who love pizza
3. Set of students who like to play basketball

EXAMPLE 5 Fill in each blank with ∈ or ∉.

Set A = { even numbers between 0∧10 }
Set B = { letters of the english alphabet between c∧ j }
1. 2 ______ A 4. b ______ B
2. 5 ______ A 5. g ______ B
3. 12 ______ A 6. e ______ B
SOLUTIONS: A={ 2 , 4 , 6 , 8 }
B= { d , e , f , g , h , i }
1. 2∈A
2. 5 ∉A, since 5 is not an even number.
3. 12 ∉A, since 12 is not between 0 and 10.
4. b ∉B, since b is not between c and j.
5. g∈B
6. e∈B

Well – defined Sets

In a set-builder notation, some situations exist that make it difficult to determine whether an object
belongs to the set or not. For instance, if we are interested in “pretty girls in the Philippines” it might be
difficult to determine if a particular girl belongs to the set. As they say, beauty is in the eye of the
beholder. Since beauty is subjective, the set “pretty girls in the Philippines” is not well-defined.

However, if we are interested in “teachers in your school” or “distinct letters in the word
HONESTY”, we know exactly if an object if an object is an element of the set or not. These sets are well-
defined.

EXAMPLE 6 State whether the collection is a well-defined set.

1. { subjects∈Grade 7 }
2. { popular actors }
SOLUTIONS: 1. Yes, because it is clear whether a subject is taught in Grade 7 or not.

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 A collection is well defined if, given an object, it can be categorically identified to belong to the collection
or not.
2. No, because some people may consider an actor popular while others may not.

1.1.3 NUMBER OF ELEMENTS IN A SET

How many days are there in a week?

How many of these days are school days?

There are 7 days in a week. If A represents the days of the

week, we denote the number of elements in A as n(A) = 7.
The answer to the question “how many?” is a cardinal number.
If B represents school days, then set B = { Monday , Tuesday , Wednesday ,Thursday , Friday }. Its cardinal
number is 5, written n(B) = 5. We say that the cardinality of set B is 5.
The cardinality of a set is the number of elements that are in a set.

ACTIVITY NO. 2
Count Me In
Stand up if you are a member of this set.
F={ students with a facebook account }
B= { students whoride a helicopter ¿ school every day }

A set with no elements is called an empty or null set. It is denoted by the symbol { }∨∅ .

ACTIVITY NO. 3
Listing the Elements of the Set
Work in pairs.
List the elements of each set.
1. The set of counting numbers less than 8
2. The set of counting numbers greater than 4
3. The set of subjects that you have in school
4. The set of stars in the sky

Sets may be described as finite or infinite.

A set is a finite set if it is empty or if it can be placed into a one-to-one correspondence with a set of
the form { 1 , 2, 3 , … , N }, where N is a counting number. In other words, the number of elements in a finite
set is a whole number. If is not finite, it is said to be infinite.
Finite set – it is a type of set wherein the number of elements is countable.
Infinite set – it is a type of set wherein the number of elements is uncountable.

EXAMPLE 7 Decide whether the set is finite or infinite.

1. the set of counting numbers greater than 10
2. the set of whole numbers less than 5
3. the set of whole numbers less than 0
SO

Example 1: Determine, with explanation, whether the given collection is well – defined or not.
1. Collection of big bags
2. Collection of leafy vegetables
3. Collection of soft stuffed toys
4. Collection of red fruits
Solution:

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 1. This collection is not well – defined because to determine whether a bag is big or not
depends on the observer.
2. This collection is well – defined because we know which vegetables are leafy and which one
are not.
3. This collection is not well – defined because we do not know how soft a stuffed toy is to
belong to the collection.
4. This collection is well – defined because we know if a fruit is red or not.

Example 2: Which of the following collections are sets and which are not? Explain your answer.
1. Collection of round objects
2. Collection of animals with tails
3. Collection of small boxes
4. Collection of Math books
5. Collection of sweet foods
Solution:
Collections mentioned in 1, 2, and 4 are sets because they are well – defined. Collections in 3 and 5
are not sets because they are not well – defined. “Small” and “sweet” are subjective modifiers which
depend on the observer.
Types of Sets
A universal set (usually denoted by U) is a set which has elements of all the related sets, without
any repetition of elements.
A set A is a subset (⊆) of another set B if all elements of the set A are elements of the set B. In
other words, the set A is contained inside the set B. The subset relationship is denoted as A⊆B.
For example, if A is the set {♢,♡,♣,♠} and B is the set {♢,△,♡,♣,♠}, then A⊆B since all the
elements of A is in set B, but B⊈A. Since B contains elements not in A, we can say that A is a proper
subset of B.
A proper subset (⊂) of a set A is a subset of A that is not equal to A. In other words, if B is a
proper subset of A, then all elements of B are in A but A contains at least one element that is not in B.
For example, if A={1,3,5} and B={1,5}then B is a proper subset of A, since all the elements in set B
are in set A. The set C={1,3,5} is a subset of A, but it is not a proper subset of A since C=A. The set D={1,4}
is not even a subset of A, since 4 is not an element of A.

Consider the sets U, A, and B.

U ={ x|x is an animal on land }
A={ dog , cat , carabao , eagle , monkey }
B= { dog , carabao , eagle }

Set U is considered a universal set. It is the set containing all elements of which all other sets are
subsets.
Set A and B are subsets of Set U because each element of set A and Set B is an element of Set U (lahat
ng element na nasa set A at B ay nasa set U).
Set B is a proper subset of Set A because Set A has elements that are not in Set B.

The set X is a subset of the set Y, written as X ⊆ Y, if every element of X is an element of Y. if

Subset of a Set

Y has at least one element that is not in X, then X is called a proper subset Y.

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Methods of Writing Sets
1. Roster Method – listing of elements inside the brackets
Example:
Write the set of numbers from 1 to 10
A = {1,2,3,4,5,6,7,8,9,10}
2. Set – builder Notation – describing the elements
x|x is read as “the set of all x's, such that x”
Example:
B={x|x is an odd number between 11 and 20} which means set B contains all the odd
numbers between 11 and 20.
By using the roster method, set B can be written as B={11,13,15,17,19}

In the following
table, each set is
written in
both the methods

If Set C is a set of fish in a set of animals on land, then Set C has no elements since there are no fish

Set C is an empty or null set denoted by the symbol { } or ∅.

living on land.

An empty or null set is always a subset of any set.

Example 3: Name three subsets of whole numbers using Roster Method and Set – builder Notation.

Solution:
Roster Method Set – builder Notation
A={ 2 , 4 , 6 , 8 ,10 } A={ x|x is an even number less than12 }
B= {1 , 3 ,5 , 7 , 9 } B= { x|x is an odd number less than10 }
C={ 5 , 10 ,15 , 20 , 25 } C={ x|x } is a multiple of 5 less than30

Example 4: List all the possible subsets of Set G= {T , E , A , M }

Solution:
{}, { T } , { E } , { A } { M }
{T , E }, {T , A }, {T , M } , {E , A }, { E , M }, { A , M }
{T , E , A} , {T , E , M }, { E , A , M }, {T , M , A }
{T , E , A , M }

Cardinality of a Set

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There are two types of set:

The cardinality of a finite set S denoted by |S| is the number of elements of the set.
The cardinality of an infinite set T is infinite and denoted by |T |=∞.

Consider Set X ={ 10 , 20 ,30 , 40 , 50 }

Y ={}
Set X has 5 elements.
Set Y has no elements.
The cardinality of set X and set Y are 5 and 0, respectively, and is denoted by
| X|=5
|Y |=0
Example 5: Which of the set containing the following elements are finite? Which are infinite? If it is finite,
give its cardinality.
1. The counting numbers less than 30.
2. The points on a given line.
3. The female presidents of the Philippines.
4. Integers greater than or equal to 13.
Solution:
1. Finite; 29 3. Finite; 2
2. Infinite 4. Infinite

LESSON 2: Set Operations

New sets can be formed from a given number of sets when operations are done on them.

Union and Intersection of Sets

Consider the sets A={ 1 ,2 , 4 ,5 } and
B= {2 , 3 ,5 ,7 , 9 } .
Union of sets is formed when you combine all the elements of 2 or more set in a single set. It is denoted
by ∪.
Example: A ∪ B read as “A union B”
Set C={ 1 , 2, 3 , 4 ,5 , 7 , 9 }
Set C is formed by combining the elements of set A and B.

Intersection of sets is formed by getting the elements common to both sets.

Example: A ∩ B read as “A intersection B”
Set D= { 2, 5 , 7 }
Set D is formed by getting the elements common to Sets A and B.

A Venn diagram can be used to represent the union and intersection of sets.

A B A B

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Example 6: Use the Venn diagram to show A ∪ B and A ∩ B. List down their elements.
A={ t , o , n , y }
B= { p , o , n , y }
Solution:

O O
t n p t n p
y y

¿ { t , o ,n , y , p } ¿ { o ,n , y }
A∪B A∩B

Complement of a Set
The shaded region of the Venn diagram represents A’.
The complement of Set A, written as A’ and as “ A complement”,
is the set of elements in the universal set U that do not belong to
set A.
If U ={ p ,q ,r , s , t } and A={ q , s } , then A' = { p , r , t } .

Example 7:
Let U ={ 0 ,1 , 2 ,3 , 4 , 5 ,6 ,7 , 8 , 9 }
A={ 1 ,3 , 5 , 6 }
B= {2 , 4 ,5 , 6 }
Find: A’, B’, (A ∪ B)’, (A ∩ B)’
Solution:
A = { 0 ,2 , 4 ,7 , 8 , 9 }
'

B ={ 0 , 1 ,3 , 7 , 8 , 9 }
'

(A ∪ B)’
A ∪ B ¿ { 1 ,2 , 3 , 4 ,5 , 6 }
(A ∪ B)’¿ { 0 , 7 , 8 , 9 }
(A ∩ B)’
A ∩ B ¿ {5 , 6}
(A ∩ B)’¿ { 0 , 1 ,2 , 3 , 4 , 7 , 8 , 9 }

Difference of Two Sets

The difference of Sets A and B (A – B) is the set whose elements belong to Set A which do not belong
to set B (lahat ng element na nasa set A na wala sa set B).

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Consider Set A={ 11,12 , 13 , 14 , 15 } and
Set B= {11, 12 ,13 ,16 , 17 }

We can find the difference of Sets A and B.

A – B = { 14 , 15 }
B – A = { 16 , 17 }

Using the Venn diagram, A – B and B – A are shown as

11 11
14 16 14 16
12 12
15 17 15 17
13 13

A–B B–A

Problems Involving Sets

We can apply set equations in solving problems.
Consider this problem:
In a class of 40, 12 students prefer taking French Language as elective subject, 15 prefer
Chinese Language, and 8 like to learn both languages.
a) How many students prefer French or Chinese language?
b) How many students prefer Chinese but not French?
c)How many students prefer only one of the two languages?
d) How many students prefer either of the two languages?
The information given in the problem can be written as
U = 40 C = 15
F = 12 F∩C=8
Using the Venn diagram,
F C U

4 8 7 40 – (8 + 4+ 7) = 21

21

We have to know first the following:

 Number of students who prefer French language only: 12 – 8 = 4.
 Number of students who prefer Chinese language only: 15 – 8 = 7.
Note: 8 is the intersection in the Venn diagram or those who wants to learn both languages.

Example 8: Loida interviews 75 students at a certain high school on the sports they are interested in. the
results are as follows: 40 play badminton, 35 play football, and 15 play both badminton and football.
1. How many students play badminton only?
2. How many students play badminton or football?

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 3. How many students do not play football?
4. How many students play neither badminton nor football?
5. How many students play exactly one of the two sports?
Solution:
B F U
n(B)= 40
n(F) = 35
n(B ∩ F) = 15

25 15 20 75 – (25+15+20) = 15

15

1. The number of students who play badminton only is 40 – 15 =25.

2. The number of students who play badminton or football is 25 + 20 + 15 = 60.
3. The number of students who do not play football is 25.
4. The number of student who plays neither badminton nor football is 15.
5. The number of students who play exactly one of the two sports is 25 + 20 = 45.

Example 9: A survey was conducted with 50 people on the ice cream flavor they like. It was found out
that 28 like chocolate, 15 like mango, and 20 like strawberry. Furthermore, 8 like chocolate and
strawberry, 7 like chocolate and mango, and 4 like all the flavors.
Find the number of people who like
a. Chocolate only
b. Strawberry only
c. Mango only
d. Chocolate and strawberry but not mango
e. None of the flavors.
Solution:
n(C) = 28 n(C ∩ S) = 8
U
C S n(M) = 15 n(M ∩ S) = 11
n(S) = 20 n(C ∩ M) = 7
17 4 5 n(C ∩ M ∩ S)= 4
4
3 7
50 – (17 + 1 + 5 + 4 + 3 + 7 + 4) = 9

M 9
a. The number of people who like chocolate only is 28 – 11 = 17.
b. The number of people who like strawberry only is 20 – 15 = 5.
c. The number of people who like mango only is 15 – 14 = 1.
d. The number of people who like chocolate and strawberry but not mango is 17 + 4 +5 = 26.
e. The number of people who like none of the flavors is 50 – (17 + 1 + 5 + 4 + 3 + 7 + 4) = 9.

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 Checked by:
Prepared by: FELICIANA K. JACALNE
DEVIE ANNE G. BISCARRA OIC Principal
Subject Teacher

Name:______________________________________
_ Week 1 & 2 Activity Score:________________
Section:________________________
I. Determine whether the given collection is well – defined or not. Write your answer on the space
provided before the number.
_______________________1. Collection of universities in the Philippines
_______________________2. Collection of beautiful paintings
_______________________3. Collection of students in a particular class
_______________________4. Collection of high chairs
_______________________5. Collection of delicious meal
_______________________6. Collection of animals with wings
_______________________7. Collection of easy physical activities
_______________________8. Collection of land transportation
_______________________9. Collection of numbers divisible by 2
_______________________10. Collection of flowers

II. Types and Cardinality of Sets

A.List all the possible subsets of Set H= { 4 , 3 , 6 , 5 }.

B. Which of the sets containing the following elements are finite? Which are infinite? If it is finite, give
its cardinality.
1. The days in a week
2. The counting numbers greater than 30
3. The letters in the English alphabet
4. The stars in the sky
5. The months in a year that begins with letter B

III. Venn Diagram

Use the Venn diagram to show R ∪ S and R ∩ S. List down their elements.
R={ 1, 2 , 3 , 4 }

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 S= { 3 , 4 , 5 ,6 }

IV. Problem Solving

A survey of 66 students showed that 23 like mango, 34 like pineapple, and 6 like both mango and
pineapple. Use Venn diagram to represent your answer.
1. How many students like mango only?
2. How many students like mango or pineapple?
3. How many students do not like pineapple?
4. How many students like neither of the two fruits?
5. How many students like only one of the two fruits?

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