Problems Involving Sets: A Mathematics Module for Grade 7, Slides of Mathematics

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Mathematics

Quarter 1 – Module 2:

Problems Involving Sets

Mathematics – Grade 7

Alternative Delivery Mode

Quarter 1 – Module 2: Problems Involving Sets

First Edition, 2020

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Published by the Department of Education

Secretary: Leonor Magtolis Briones

Undersecretary: Diosdado M. San Antonio

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Office Address: Regional Center, Brgy. Carpenter Hill, City of Koronadal

Telefax: (083) 2288825/ (083) 2281893

E-mail Address: [email protected]

Development Team of the Module

Writers: Joy D. Padernal , Isabelita D. Tenorio, Joven V. Felongco, Josephine G. Tibay

Editors: Randy L. Pendilla, Raul Pojas, Mary Jean Nequinto

Reviewers: Evelyn C. Frusa PhD, Noemi E. Parcon, Rolex H. Lotilla, Arvin M. Tejada

Illustrator: None

Layout Artist: Iza May S. Agrazamendez

Management Team: Dr. Allan G. Farnazo, CESO IV - Regional Director

Gilbert B. Barrera – Chief, CLMD

Arturo D. Tingson, Jr. – REPS, LRMS

Peter Van C. Ang-ug – REPS, ADM

Jade T. Palomar – REPS, Mathematics

Belen L. Fajemolin , PhD - CID Chief

Evelyn C. Frusa, PhD – EPS – LRMS

Bernardita M. Villano – ADM Coordinator

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Introductory Message

For the facilitator:

Welcome to the Mathematics 7 Alternative Delivery Mode (ADM) Module on Problems

Involving Sets!

This module was collaboratively designed, developed and reviewed by educators both

from public and private institutions to assist you, the teacher or facilitator in helping

the learners meet the standards set by the K to 12 Curriculum while overcoming

their personal, social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and independent

learning activities at their own pace and time. Furthermore, this also aims to help

learners acquire the needed 21st century skills while taking into consideration their

needs and circumstances.

In addition to the material in the main text, you will also see this box in the body of

the module:

As a facilitator you are expected to orient the learners on how to use this module.

You also need to keep track of the learners' progress while allowing them to manage

their own learning. Furthermore, you are expected to encourage and assist the

learners as they do the tasks included in the module.

Notes to the Teacher

This contains helpful tips or strategies that

will help you in guiding the learners.

iii

For the learner:

Welcome to the Mathematics 7 Alternative Delivery Mode (ADM) Module on

Problems Involving Sets!

The hand is one of the most symbolized part of the human body. It is often used to

depict skill, action and purpose. Through our hands we may learn, create and

accomplish. Hence, the hand in this learning resource signifies that you as a learner

is capable and empowered to successfully achieve the relevant competencies and

skills at your own pace and time. Your academic success lies in your own hands!

This module was designed to provide you with fun and meaningful opportunities for

guided and independent learning at your own pace and time. You will be enabled to

process the contents of the learning resource while being an active learner.

This module has the following parts and corresponding icons:

What I Need to Know

This will give you an idea of the skills or

competencies you are expected to learn in the

module.

What I Know

This part includes an activity that aims to

check what you already know about the

lesson to take. If you get all the answers

correct (100%), you may decide to skip this

module.

What’s In

This is a brief drill or review to help you link

the current lesson with the previous one.

What’s New

In this portion, the new lesson will be

introduced to you in various ways such as a

story, a song, a poem, a problem opener, an

activity or a situation.

What is It

This section provides a brief discussion of the

lesson. This aims to help you discover and

understand new concepts and skills.

What’s More

This comprises activities for independent

practice to solidify your understanding and

skills of the topic. You may check the

answers to the exercises using the Answer

Key at the end of the module.

What I Have Learned

This includes questions or blank

sentence/paragraph to be filled in to process

what you learned from the lesson.

What I Need to Know

This module was designed and written with you in mind. It is here to help you

master your skills in solving mathematical problems involving sets. The scope of this

module permits it to be used in many different learning situations. The language

used recognizes the diverse vocabulary level of students. The lessons are arranged

to follow the standard sequence of the course. But the order in which you read them

can be changed to correspond with the textbook you are now using.

The module is all about Solving Problems Involving Sets.

After going through this module, you are expected to:

1. solve problems involving sets using Venn diagram;
2. apply set operations to solve a variety of word problems.

What I Know

Read each question below. You may draw a Venn diagram to help you find the

answer. Select your answer from the choices lettered A to D and write the letter of

your choice on a separate sheet of paper.

For items 1 – 2.

Let A = { 1, 2, 3, 4, 5, 6 } , B = { 2, 4, 5, 6 }

1. Find A U B.

A. { 1, 2, 3, 4, 5, 6 } B. { 2, 4, 5, 6 } C. { 2, 4, 6} D. { 1, 3, 5 }

2. Find A ∩ B.

A. { 1, 2, 3, 4, 5, 6 } B. { 2, 4, 5, 6 } C. { 2, 4, 6} D. { 1, 3, 5 }

For items 3 - 4. Given the Venn Diagram below.

3. Find M U S.

A. { 5, 10, 15 } B. { 10, 15 } C. { 5, 10 } D. { 15 }

4. Find M ∩ S.

A. { 5, 10, 15 } B. { 10, 15 } C. { 5, 10 } D. { 15 }

5. In the Venn diagram below, which of the following is represented by the shaded

region?

A. X U Y B. X ∩ Y C. X ‘ D. all of the above

6. Which of the following is represented by the Venn diagram below?

A. A – B B. B – A C. A U B D. A ∩ B

X Y

U

U

S M

A B

u

Try solving the following problem:

In a group of 120 students, 68 had ridden a bus, 78 had ridden the LRT, 33

had ridden a jeep, while 40 had ridden both the bus and the LRT , 20 had ridden

the bus and the jeep,19 had ridden the LRT and the jeep and 15 had ridden the

bus, the LRT and the jeep.

a. How many had ridden the bus only?

b. How many had ridden the LRT only?

c. How many had ridden the jeep only?

d. How many did not ride on any of the three modes of transportation?

Lesson

1

Problems Involving Sets

What’s In

Notes to the Learner

Find the solution using any method.

What’s New

Venn diagram is a principal way of showing sets diagrammatically. This

method consists primarily of entering the elements of a set into a circle or circles. It

can be used to solve word problems involving union and intersection of sets.

In solving set operations using the Venn diagram, the following are the steps

to be followed:

Step 1. Determine what is given and what are being asked.

Step 2. Illustrate using the Venn diagram.

Step 3. Determine what operations to be used.

Step 4. Use the operations.

Step 5. Answer the questions being asked.

Here are some worked out examples:

Example 1.

Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(AUB)=36.

Find n(A∩B).

Solution:

Step 1. Determine what is given and what are being asked.

Given :

n(A) = 20,

n(B) = 28

n( A U B ) = 36.

Asked:

Find n( A∩ B).

Step 2. Illustrate using the Venn diagram if possible.

The Venn diagram is shown below

A A∩B B

U

Asked:

a. How many students liked only tea?

b. How many students liked only coffee?

c. How many students liked neither tea nor coffee?

Step 2. Illustrate using the Venn diagram.

Let T = set of students who like only tea

C = set of students who liked only coffee

X = set of students who liked neither tea nor coffee

The Venn diagram is shown below

Step 3. Determine what operations to be used.

(1)To obtain T,

T = 140 - 80 students who liked tea minus students who

liked both tea and coffee

T = 60 set of students who liked only tea

(2) To obtain C,

C = 120 – 80 students who liked coffee minus students

who liked both tea and coffee

C = 40 set of students who liked coffee only

(3) To obtain X,

X = 200 – ( T + C + 80) total number of students who were

randomly selected minus the sums of T , C

and 80

X = 200 – ( 60 + 40 + 80) by substitution

X = 200 – 180 by simplifying

X = 20 set of students who liked neither tea nor

coffee

C

T 80

U

X

Step 4. Use the operations.

The number of elements in each region is shown below:

Step 5. Answer the questions being asked.

a. How many students liked only tea? 60 students

b. How many students liked only coffee? 40 students

e. How many students liked neither tea nor coffee? 20 students

Example 3.

A group of 25 high school students were asked whether they used either

Facebook or Twitter or both. Fifteen (15) of these students used Facebook,

and twelve (12) used Twitter.

a. How many students used Facebook only?

b. How many students used Twitter only?

c. How many students used both Social networking sites?

Solution:

Step 1. Determine what is given and what are being asked.

Given:

25 high school students who were asked whether they use either

Facebook or Twitter or both

15 students who used Facebook

12 students who used Twitter

Asked:

a. How many used Facebook only?

b. How many used Twitter only?

c. How many used both Social networking sites?

Step 2. Illustrate using the Venn diagram.

Let F = set of students who used Facebook only

T = set of students who used Twitter only

B = set of students who used both social networking sites

U

Step 5. Answer the questions being asked.

a. How many used Facebook only? 13 students

b. How many used Twitter only? 10 students

c. How many used both Social networking sites? 2 students

Example 2.

A group of 50 students went for a tour in South Cotabato province. Out

of 50 students, 24 joined the trip in Lake Sebu for a zipline experience, 18

went to the flower farm in Tupi, 20 went to Si – ok falls in Koronadal City, 12

joined the trip to Lake Sebu and Tupi, 15 went to Tupi and Si-ok falls and 11

made a trip to Lake Sebu and Si-ok falls and 10 visited the three tourists

spots.

a. How many of the students went to Lake Sebu only?

b. How many of the students went to Tupi only?

c. How many joined the Si-ok trip in Koronadal City only?

d. How many did not go to any of the tourist spots?

Solution:

Step 1. Determine what is given and what are being asked.

Given:

50 students went for a tour

24 students who visited Lake Sebu

18 students who went to Tupi

20 students who went to Koronadal City

12 students who joined the trip to Lake Sebu and Tupi

15 students who went to Tupi and Koronadal City

11 students who went to Lake Sebu and Koronadal City

10 students who visited the three tourist spots

Asked:

a. How many of the students went to Lake Sebu only?

b. How many of the students went to Tupi only?

c. How many joined the Si-ok trip in Koronadal City only?

d. How many did not go to any of the tourist spots?

Step 2. Illustrate using the Venn diagram.

We will let,

LKT = number of students who visited the THREE tourist spots

L = number of students who visited Lake Sebu only

T = number of students who went to Tupi only

K = number of students who went to Koronadal City only

LT = number of students who joined the trip to Lake Sebu and Tupi only

TK = number of students who went to Tupi and Koronadal City only

LK = number of students who went to Lake Sebu and Koronadal City only

N = number of students who DID NOT see any of the THREE tourist spots

Step 3 : Determine what operations to be used.

(1) To obtain LKT,

LKT = 10 students who visited the THREE tourist

spots

(2) To obtain TK,

TK = 15 – LKT students who went to Tupi and Koronadal

Cityminus LKT

TK = 15 – 10 by substitution

TK = 5 students who went to Tupi and Koronadal

City only

(3) To obtain LT ,

LT = 12 – LKT students who went to Lake Sebu and Tupi

minus LKT

LT = 12 - 10 by substitution

LT = 2 students who joined the trip to Lake Sebu

and Tupi only

(4) To obtain LK,

LK = 11 – LKT students who went to Lake Sebu and

Koronadal City minus LKT

LK = 11 – 10 by substitution

LK = 1 students who went to Lake Sebu and

Koronadal City only

( 5) To obtain L ,

L = 24 – (LKT + LT + LK) students who joined the trip to Lake Sebu

minus the sums of LKT, LT and LK)

L = 24 – (10 + 2 + 1) by substitution

L = 24 – 13 by simplying

L = 11 students who visited Lake Sebu only

LT

LK

LKT

LT

TK

N

L

K

T

U

What is It

In a group of 120 students, 68 had ridden a bus, 78 had ridden the LRT,

33 had ridden a jeep, while 40 had ridden both the bus and the LRT, 20 had

ridden the bus and the jeep,19 had ridden the LRT and the jeep and 15 had

ridden the bus, the LRT and the jeep.

a. How many had ridden the bus only?

b. How many had ridden the LRT only?

c. How many had ridden the jeep only?

d. How many did not ride on any of the three modes of transportation?

Solution:

LRT

Bus

Jeep

U

Can you explain the numbers?

Do the following. Represent the sets and draw a Venn Diagram.

1. Among the 70 residents in Barangay General P. Santos, 53 liked eating in

Restaurant A while 42 liked eating in Restaurant B. How many liked

eating both in Restaurant A and Restaurant B? In Restaurant A only? In

Restaurant B only?

2. The following diagram shows how all the Grade Seven students of

Koronadal National Comprehensive High School go to school.

a. How many students ride in a car, tricycle and the motorcycle in going

to school?

b. How many students ride both in a car and a tricycle?

c. How many students ride both in a car and the motorcycle?

d. How many students ride both in a tricycle and the motorcycle?

e. How many students go to school in a car only? Tricycle only? in the

motorcycle only? walking?

f. How many Grade Seven students of Koronadal National

Comprehensive High School are there in all?

What’s More

Walking

Tricycle

Motorcycle

Car