Table of C ntents - mysapbooks.com fileChapter Test This is a summative test given at the end in...

Synergy for Success in Mathematics 7 is designed for Grade 7 students. The textbook contains all the required learning competencies and is supplemented with some additional topics for enrichment. Lessons are presented using effective Singapore Math strategies that are intended for easy understanding and grasp of ideas for its target readers. Various exercises are also provided to help the learners acquire the necessary skills needed.

The book is organized with the following recurring features in every chapter:

Learning Goals Thisgivesthespecificobjectivesthatareintendedtobeachieved in the end.

Introduction The reader is given a bird's eye view of the contents.Historical Note A brief historical account of a related topic is included giving the

reader an awareness of some important contributions of some great mathematicians or even stories of great achievements related to mathematics.

Method/Exam Notes These additional tools help students recall important information, formulas, and shortcuts, needed in working out solutions.

Examples Step-by-stepanddetaileddemonstrationsofhowaspecificconcept or technique is applied in solving problems.

Enhancing Skills These are practice exercises found after every lesson, that will consolidate and reinforce what the students have learned.

Linking Together This visual tool can help the students realize the connection of all the ideas presented in the chapter.

Chapter Test This is a summative test given at the end in preparation for the expected actual classroom examination containing the topics included in the chapter. A challenging task is designed for the learner giving him an opportunity to use what he/she has learned in the chapter.

Chapter Project Thismaybeamanipulativetypeofactivitythatisspecificallychosen to enhance understanding of the concepts learned in the chapter.

Making Connection The students are exposed to facts and information that connect mathematics and culture. This is for the purpose of letting the learnersappreciatethesubjectbecauseoftangibleortrue-to-life stories that show how mathematics is useful and relevant.

Every effort has been made in order for all the discussions in this book to be clear, simple, and straightforward. This book also gives opportunities for the readers to see the beauty of mathematics as an essential tool in understanding the world we live in. With this in mind, appreciation of mathematics goes beyond seeing; realizing its critical application to decision making in life completes the purpose of knowing and understanding mathematics.

PREFACE

Table of C ntents

Introduction ............................................................................................................................1Historical Note .......................................................................................................................21.1 IntroductiontoSets ...............................................................................................31.2 OperationsonSets ............................................................................................... 14Linking Together ................................................................................................................ 24Chapter Test ........................................................................................................................ 25ChapterProject ................................................................................................................... 29Making Connection ........................................................................................................... 30

Introduction ......................................................................................................................... 31Historical Note .................................................................................................................... 322.1 TheRealNumberSystem .................................................................................. 332.2 PropertiesofRealNumbers ............................................................................. 392.3 Integers ..................................................................................................................... 462.4 AbsoluteValueofaRealNumbers ............................................................... 642.5 Fractions ................................................................................................................... 682.6 Decimals ................................................................................................................... 822.7 ApproximationonSquareRoots .................................................................... 912.8 ScientificNotation ................................................................................................ 962.9 SignificantDigits .................................................................................................101Linking Together ..............................................................................................................105Chapter Test ......................................................................................................................106ChapterProject .................................................................................................................109Making Connection .........................................................................................................110

CHAPTER 1 BASIC IDEA OF SETS

CHAPTER 2 REAL NUMBERS

Introduction .......................................................................................................................111Historical Note ..................................................................................................................1123.1 MeasuringLength,Perimeter,Mass,andVolume .................................1133.2 MeasuringArea,Temperature,andTime.................................................131Linking Together ..............................................................................................................145Chapter Test ......................................................................................................................146ChapterProject .................................................................................................................149Making Connection .........................................................................................................150

Introduction .......................................................................................................................151Historical Note ..................................................................................................................1524.1 NumberSequenceandPatternFinding ....................................................1534.2 AlgebraicExpressions .....................................................................................1594.3 IntegralExponents .............................................................................................1704.4 EvaluationofAlgebraicExpressions .........................................................1864.5 TranslationofMathematicalPhrasesintoSymbols ............................1944.6 OperationsofPolynomials .............................................................................2024.7 SpecialProducts ..................................................................................................215Linking Together ..............................................................................................................227Chapter Test ......................................................................................................................228ChapterProject .................................................................................................................231Making Connection .........................................................................................................232

Introduction .......................................................................................................................233Historical Note ...................................................................................................................2345.1 LinearEquationsinOneVariable ................................................................2355.2 SolvingAbsoluteValueEquations ...............................................................2545.3 LinearInequalitiesinOneVariable ............................................................2595.4 SolvingAbsoluteValueInequalities ...........................................................2785.5 SolvingWordProblemsInvolving Linear Equations and Inequalities ..............................................................286Linking Together ..............................................................................................................310Chapter Test ......................................................................................................................311ChapterProject .................................................................................................................315Making Connection .........................................................................................................316

CHAPTER 3 MEASUREMENT

CHAPTER 4 ALGEBRAIC EXPRESSIONS

CHAPTER 5 SIMILARITY AND RIGHT TRIANGLES

Introduction .......................................................................................................................317Historical Note ...................................................................................................................3186.1 ObjectsofGeometry ..........................................................................................3196.2 AnglesandAngleMeasures ...........................................................................3376.3 ReasoningandProving ....................................................................................3536.4 MoreObjectsofGeometry ..............................................................................3666.5 GeometricConstructions ................................................................................378Linking Together ..............................................................................................................385Chapter Test ......................................................................................................................386ChapterProject .................................................................................................................389Making Connection .........................................................................................................390

Introduction .......................................................................................................................391Historical Note ...................................................................................................................3927.1 PerpendicularandParallelLines .................................................................3937.2 ApplyingConceptsofPerpendicularandParallelLines ....................405Linking Together ..............................................................................................................414Chapter Test ......................................................................................................................415ChapterProject .................................................................................................................419Making Connection .........................................................................................................420

Introduction .......................................................................................................................421Historical Note ...................................................................................................................4228.1 IntroductiontoStatistics .................................................................................4238.2 TheFrequencyTable .........................................................................................4298.3 UseofGraphstoRepresentandAnalyzeData .......................................4388.4 MeasuresofCentralTendency(UngroupedData) ...............................447Linking Together ..............................................................................................................458Chapter Test ......................................................................................................................459ChapterProject .................................................................................................................463Making Connection .........................................................................................................464

Glossary .....................................................................................................................................465 Index ...........................................................................................................................................479 Bibliography ............................................................................................................................486

CHAPTER 6 TOOLS OF GEOMETRY

CHAPTER 7 PERPENDICULAR AND PARALLEL LINES

CHAPTER 8 STATISTICS

BASIC IDEA OF SETS1

Our daily activities often involve groups or collection of objects, such as set of wardrobe, group of students, a collection of toys, a list of formulas, and many others.

One of the important foundation for some topics in mathematics is the idea of sets. This chapter covers the fundamental concepts of a set, kinds of sets, union of sets, and intersection of sets.

Learning GoalsAt the end of the chapter, the students should be able to:

1.1 Defineanddescribeasetand use a Venn diagram to illustrate a set and properties of set operations

1.2 Describe and illustrate complement of a set, and union and intersection of sets

2

George Ferdinand Ludwig Philipp Cantor (1845–1918) is a German mathematician known as the founder of set theory. Cantorsetforththemoderntheoryoninfinitesetsthatdevelopedallthedisciplinesinmathematics.Cantordefinedwell-orderedand infinite sets. He established the importance of one-to-one correspondence between the members of two sets. He showedthatnotallinfinitesetshavethesamesize,therefore,infinitesetscanbecomparedwithoneanother.Hethenprovedthatthereal numbers are “numerous” than the natural numbers.He definedwhat it means for two sets to have the same cardinal number. Heprovedthatthesetofrealnumbersandthesetofpointsinn- dimensional Euclidean space have the same exponent.

Cantor’s early interests were in number theory, indeterminate equations, and trigonometric series. In 1874, he started his radical workonsettheoryandthetheoryoftheinfinite.Cantorcreatedawholenewfieldofmathematicalresearch.

Historical Note

3

Synergy for Success in Mathematics Chapter 1

A set is a collection of objectswhich are clearly defined as belongingtoawell-definedgroup.Eachobjectinasetiscalledan element of a set. Each element is separated by a comma. The set is enclosed by braces { }. Normally, a capital letter is used to name or label a set.

For example, set A consists of all subjects offered in secondary school.

A = {set of subjects in secondary school}

A = {English, Math, Science, CLE, Filipino, Social Studies, MAPEH}

Asetmustbewelldefinedsothatwecandeterminewhetheran object is an element of the set.

A set may be described using a set notation. The two main methods of set notation are the rule method or set builder notation and the roster or listing method.

Rule Method Roster or Listing Method

A x x={ }: is a counting number from 1 to 5 A={ }1 2 3 4 5, , , ,

B x x={ }: is a month that starts with letter A B={ }April, August

C x x={ }: is a prime factor of 15 C={ }3 5,

In roster or listing method, the elements are separated by commas and are enclosed within a pair of brace { }.

1.1 Introduction to Sets

4

Notice that set A in the rule method is properly described so that it could be easier to list down all the possible elements.

A x x={ }: is a counting number from 1 to 5 is read as “A is the set of elements x, such that x is a counting number from 1 to 5.”

There are cases when it is too tedious or impossible to list all the elements of a set. There are sets whose elements are infiniteortoomanytoencloseinsidebraces.Suchsetsareratherdefinedusingtherulemethod.

For example:

A x x={ }: is an even number between 1 and 100

A={ }2 4 6 8 96 98, , , , , ,

The three dots (...) are called ellipsis which means "continue on." The ellipsis represents the other elements which are no longer practical to include in the list.

List all the elements of the following sets.

(a) A = {x : x is a letter in the word SUBTRACT}(b) B = {x : x is a counting number greater than 8}

SOLUTION

(a) A = {S, U, B, T, R, A, C} Although there are two T's, this letter must be

written only once within the brace.(b) B = {9, 10, 11, 12, ...} The ellipsis is used to acknowledge the existence of

other elements. It indicates that there are infinitecounting numbers greater than 8, which is impossible to list them all down.

Example 1

5


The table below indicates the common symbols used to show the relationship between sets and elements.

Symbol WordsÎ elementÏ not an elementÌ subset; part ofË not a subset; not a part ofÆ empty; no element; null setÈ union; combine elementsÇ intersection; common element(s)

To relate or describe the relationship between an element and a set, we use Îand Ï .

For example: If A = {a, e, i, o, u}, then u and b∈ ∉A A.

This implies that “u belongs to A” and “b is not an element of A.”

Given: B={ }all the factors of 24

Fill in the blanks with ∈ ∉ or .

(a) 1 B(b) 15 B(c) 8 B(d) 4 B(e) 12 B(f) 16 B

SOLUTION

(a) 1ÎB(b) 15Ï B(c) 8ÎB(d) 4ÎB(e) 12ÎB(f) 16ÎB

Example 2

6

The factors 1, 2, 3, 4, 6, 8, and 12 are numbers which can exactly divide 24. Thus, these numbers are considered factors of 24.

The numbers 1, 2, 3, 4, 6, 8, 12, and 24 can exactly divide 24. Thus, these numbers are considered factors of 24.

The numbers 15 and 16 are not factors of 24 because of the existence of a remainder when 24 is divided by either of these two numbers.

Universal Set

A set that contains everything or all elements under consideration and are relevant to the problem is called a universal set, denoted as U.

A universal set could be drawn (usually as a rectangle) to contain all the members which are considered.

For example:U = {set of whole numbers less than 10}U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

1 2

5

3

6

8

4

7

9

U

7


Empty or Null Sets

A set with no elements in it is known as an empty set or null set. It is represented by ∅ { } or by (a set with no elements).However,itisneverrepresentedby ∅{ }.

For example:E = {the month of the year with more than 31 days}E = { } or E=∅ since there are no months with more than

31 days.

Determine whether each of the following sets is empty or not.(a) P = { x : x is kind of triangle having sides of

different lengths} (b) Q = {x : x is a factor of 16 and 20 30< < }x(c) R = {x : x is a prime number and 8 10< < }x

SOLUTION

(a) A scalene triangle has sides of different lengths. Hence,P≠∅.

(b) The factors of 16 are 1, 2, 4, 8, and 16. There are no factorsof16between20and30.Hence,Q≠∅.

(c) A prime number has only two factors, itself and 1. 9 is between 8 and 10. 9 has three factors: 1, 3, and 9.Hence,R≠∅.

Example 3

8

Subset

A subset is a portion of a set. A set is a subset of another set if and only if all the elements of a set are contained in another set.

Set Q is a subset of set P if every element of set Q is also an element of set P. If set Q is a subset of P, but not equal to set P, then Q is a proper subset of P.

Notation: Q P x Q x P⊂ ∈ ∈, if , then .

The following generalizations are consequences of thedefinition.

(1) Every set is a subset of itself. Notation: A AÌ

(2) An empty set is a subset of every set. Notation: ∅⊂ A

Fill in each of the following blanks with the symbol Ì or Ë.

(a) {6, 7, 8} ____ {0, 1, 4, 5, 6, 7, 8}(b) { j, l, q} ____ {vowels}(c) {blue, red} ____ {rainbow colors}(d) {8, 16} ____ {multiples of 16}

SOLUTION

(a) {6, 7, 8} {0, 1, 4, 5, 6, 7, 8}

6, 7, and 8 can be found in {0, 1, 4, 5, 6, 7, 8}.(b) { j,l,q} {vowels}

j, l, and q are not vowels.(c) {blue, red} {rainbow colors}

Blue and red are two of the rainbow colors.(d) {8, 16} {multiples of 16}

8 is not a multiple of 16.

Example 4

Ì

Ë

Ì

Ë

9


The number of subsets of a certain set is 2n, where n is the number of elements in the set.

If A={ }1 2 3, , , then A has 8 subsets.

Number of subsets = =2 83 , where the exponent 3 is the number of elements of A.

Hereisacompletelistofthe8subsets. Improper subset: 1 2 3, ,{ } Proper subset with two elements: 1 2 1 3

2 3

, , , ,

,

{ } { }{ }

Proper subset with one element: 1 2 3{ } { } { }, ,

Improper subset with no element: { }

Determine the number of subsets for each of the following sets. Then, list all the subsets.

(a) D={ }7 9,

(b) E={ }p(c) F ={ }a,e,i,o

SOLUTION

(a) Number of subsets ==

2

4

2

Subsets of D: { }, {7}, {9}, {7, 9} (b) Number of subsets =

=2

2

1

Subsets of E: { }, {p}

(c) Number of subsets ==

2

16

4

Subsets of F: { }, {a, e, i, o}, {a}, {e}, {i}, {o}, {a, e}, {a, i}, {a, o}, {e, i }, {e, o}, {i, o} {a, e, i }, {a, e, o}, {e, i, o}, {a, i, o}

Example 5

10

Finite or Infinite Set

Setshavingfiniteorexact listofelementsarecalledfinite sets.Fora long list, adefinitionora setbuilderhas tobeused. If the list is short like the one shown below, it could be simply described by listing all its members.

For example:P = {set of two-digit positive integers ending with the

digit 9}P = {19, 29, 39, 49, 59, 69, 79, 89, 99}The number of elements in a finite set is denoted byP P. The symbol is read as “cardinality of set P.”

Therearesituationswherethelistcouldbeinfinite.Asetisclassifiedasinfinite when its elements cannot be counted. For example, the set of even numbers starting from 0 and continuingindefinitelyhastobestatedas

N = {x : x is an even number}

which means the list 0, 2, 4, 6, 8, ... continues indefinitely.

Given: B = {x : xisaletterinthewordMATHEMATICS} C = {x : x is a factor of 21} D = {x : x is an integer between 4 and 5}(a) List all the elements of sets B, C, and D.(b) Find B C D, , . and

SOLUTION

(a) B = {A, C, E, I, M, S, T,H} Although M, A, and T appear more than once in the

word MATHEMATICS, these three letters must be written only once inside the brace.

C={ }1 3 7 21, , , These four elements of set C are numbers which

can equally divide 21.

D D={ } =∅ or , since all the numbers between 4 and 5 are non-integers or fractions.

Example 6

11


(b) B

C

D

=

=

=

8

4

0

Special Sets

A subset is a set contained within another set, or it can be the entire set itself. The set {1, 2} is a subset of the set {1, 2, 3}, and the set {1, 2, 3} is a subset of the set {1, 2, 3}. When the subset is missing some elements that are in the set it is being compared to, it is a proper subset. When the subset is the set itself, it is an improper subset.

The symbol used to indicate “is a proper subset of” is Ì . When there is the possibility of using an improper subset, the symbol used is Í . Therefore, 1 2 1 2 3, , ,{ }⊂{ } and 1 2 3 1 2 3, , , , .{ }⊆{ } The universal set is the general category set, or the set of all those elements under consideration. The empty set, or null set, is the set with no elements or members.

Both the universal set and the empty set are subsets of every set.

12

Equal Sets and Equivalent Sets

Two sets may contain exactly the same elements or the same number of elements.

Two sets are considered equal only if each member of one set is also a member of the other, in which case it can be stated that A B= .

Two sets are considered equivalent if they contain the same number of elements.

Consider set A B={ } ={ }2 5 1 1 2 5, , , , . and

Since A and B have exactly the same elements, then A B= .They are also equivalent sets since they contain the same number of elements. This is stated as A B= .

Determine whether the following pairs of sets are equal or equivalent.

(a) X Y={ } ={ }4 5 6 7 5 7 4 6, , , ; , , ,

(b) A={common multiples of 5 and 7

which are less than 880} B={ }35 75,

(c) C D={ } ={ }0 ;

SOLUTION

(a) X Y

X Y X Y

={ } ={ }={ }= =

4 5 6 7 5 7 4 6 4 5 6 7, , , ; , , , , , ,

.So, and

(b) A B

A B A B

={ } ={ }= ≠

35 70 35 75, ; ,

,

So, A is equivalent to B but not equal.

(c) C DC D≠

= =1 0 and

So, C and D are neither equal nor equivalent.

Example 7

Method NoteThe order of the elements is not important.Equal sets are equivalent sets, but not all equivalent sets are equal sets.

13


ENHANCING SKILLS

A Write the following sets in rule method.(1) G = {x : x is a letter in the word ALGEBRA} (2) B = {x : x is a positive integer divisible by 2 or 3}(3) C = {x : x is an integer} (4) D = {x : x is a multiple of 2 and 3 between 20 and 40} (5) E = {x : x is a reciprocal of 0}

B Giventhefollowingsets,fillineachblankwithÎorÏ. F = {x : x is a positive integer divisible by 2 or 3} D = {x : x is a multiple of 2 and 3 between 20 and 40}

(6) 5 F(7) 15 F(8) 20 F(9) 36 D(10) 39 D

C Check ()theclassificationofthesetorfollowingsets.Refertothesetsinpart A.

Set(s) Finite Infinite Empty

B

C

D

E

14

Venn Diagrams

Venn diagrams are schematic diagrams used to depict collections of sets and represent their relationships. Any closed geometrical shapes (circles, ovals, rectangles, ...) can be used to represent Venn diagram.

Consider set A is specified after a universal set has been defined. Set A has to be located within the universal set as it must be a subset of the universal set. Set A could be drawn as any plane figure within the universal set.

U A

1.2 Operations on Sets

15


Set Notation Venn Diagram

A = {a, e, i, o, u}

U Aae

io

u

B = {Albert, Alex, Anne}

U BAlbert

AlexAnne

C = {4, 5, 6, 7}

D = {4, 6, 8}

U C D

5 47 6 8

E = {2, 4}

F = {1, 2, 3, 4}

U F

12

34

E

16

Example 1

Construct a Venn diagram to represent the following sets. Given U as the universal set.

(a) A = {multiples of 4 which are less than 25}

(b) B = {x : x is an integer, 5 9≤ < }x(c) C = {two-digitnumbersthatendwiththedigit1}

(d) D = {letters in the word ARITHMETIC}

(e) E = {x : x is an even number, 21 30< < }x

SOLUTION

(a) U A20

24

124168

(d) U D

R C HT M

A E I

(b) U B

6587

(e) U E22

26

2824

(c) U C11 31

71

51 812161 9141

17


Draw a Venn diagram to illustrate the relationship between each pair of sets.

(a) U x x

Z x x

={ }={ }

:

:

is a rational number

is an integer

(b) U x x

D y y y

={ }= < <

:

:

is a factor of 20

is a factor of 10 and 0 122

5 10

{ }={ }F ,

(c) U

J

K y y

={ }={ }=

2 3 4 5 6 7 8 9 10 11

3 5 9 11

, , , , , , , , ,

, , ,

: is an integer, 22 9< <{ }y

SOLUTION

(a) U Z

integers

(b) U DF

5 10

14 202

(c) U J K

9

1110 2

35

4

76

8

Example 2

18

Complement of a Set

The complement of set A with respect to the universal set U, denoted by Ac, is the set of all points that are in U but not in A.

Consider universal set U={ }1 2 3 4 5 6, , , , , and set A={ }1 2, ,

then the complement of A is given by AC ={ }3 4 5 6, , , .

U A

15

6

3

42

Ac

Union of Sets and Intersection of Sets

The operations on sets somewhat behave in a similar manner to the basic operations on numbers.

The union of a set is the result of adding or combining the elements of two or more sets.

The union of set A and set B, denoted by A È B and read as “A union B,” is the set of all elements belonging to either of the sets. Each element of the union is an element of either set A and/or set B.

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 3, 4, 5}, and B = {4, 5, 6, 7}, then A È B = {1, 2, 3, 4, 5, 6, 7}.

U A B

9

12

3

4 6

5 7

810

Exam Note

A Ì (A È B)B Ì (A È B)

19


Example 3

Determine the union of the following sets and represent it using a Venn diagram.

(a) M = {-5, -4, -3, -2} N = {-2, -1, 0, 1, 2}

(b) P = {x : x is a factor of 20}

Q = {x : x is a factor of 36}

R = {x : x is a factor of 9}

SOLUTION

(a) M È N = {-5, -4, -3, -2, -1, 0, 1, 2}

U MN

-3

-4-5

-2

-101

2

shaded region = M È N

(b) P = {1, 2, 4, 5, 10, 20}

Q = {1, 2, 3, 4, 6, 9, 12, 18, 36}

R = {1, 3, 9}

P È Q È R = {1, 2, 3, 4, 5, 6, 9, 10, 12, 18, 20, 36}

510 2

13

9

366

1215

420

UP Q

R

shaded region = P È Q È R

Method NoteArrange the elements in increasing order.

Exam Note

-2 is a common element.It must be written once only in the union of M and N.

20

The intersection of sets P and Q, denoted by P Ç Q, is the set of elements which are common to both sets P and Q.

If P = {1, 2, 3, 4, 5} and Q = {2, 4, 6, 8}, then P Ç Q = {2, 4}.

U P Q

35

1 2

4

8

6

Notice that (P Ç Q) Ì P and (P Ç Q) Ì Q.Sets with no common elements are disjoint sets.

(a) A

BA B

={ }={ }

∩

2 5 7 11 13 15 19

1 4 7 8 11 14 17 19

, , , , , ,

, , , , , , ,

.Find

(b) P

Q

R

={ }={ }=

a, c, d, f, h, j

e, g, h, i, j, k

c, f, h, i, l, m{{ }∩ ∩Find P Q R.

SOLUTION

(a) A

B

A B

={ }={ }

∩ ={ }

2 5 7 11 13 15 19

1 4 7 8 11 14 17 19

7 11 19

, , , , , ,

, , , , , , ,

, ,

(b) P

Q

R

={ }={ }=

a, c, d, f, h, j

e, g, h, i, j, k

c, f, h, i, l, m{{ }∩ ∩ ={ }P Q R h

Example 4

Exam Note

The intersection of sets A and B are the common elements in both sets.

Exam Note

The intersection of sets P, Q, and R is the common elements in the three sets.

21


Write down the intersection of the following sets and represent the intersection using a Venn diagram.

C=

1

4

2

3

3

5

7

9

9

11, , , ,

D=

1

3

2

3

3

8

4

7

9

11

10

13, , , , ,

SOLUTION

C=

1

4

2

3

3

5

7

9

9

11, , , ,

D=

1

3

2

3

3

8

4

7

9

11

10

13, , , , ,

UC D

C Ç D

1

4

3

5

7

9

2

3

9

11

1

3

4

7

3

8

10

13

C D∩ =

2

3

9

11,

Method NoteThe region where both sets overlap represents the intersection between the two sets.

Example 5

22

Write down the intersection of the following sets and represent the intersection using a Venn diagram.

F z z z

G y y

= < <{ }=

:

:

is a multiple of 5 and 10

is an even num

50

bber and 18 < <{ }={ }

y

H

42

25 27 30 33 34 40 41, , , , , ,

SOLUTION

F

G

={ }=

15 20 25 30 35 40 45

20 22 24 26 28 30 32 34 36 38 40

, , , , , ,

, , , , , , , , , ,{{ }={ }H 25 27 30 33 34 40 41, , , , , ,

UF 15

35

45 25 34

22 24283632

38

26

2741

33

2030

40

F Ç G Ç H

G

H

F G H∩ ∩ ={ }30 40,

Example 6

Method Note

The region where all the three sets overlap represents the intersection of the three sets.

23


ENHANCING SKILLS

A Answer the following using the given sets. Indicate each solution in a Venn diagram.

Given: U

A x x

= <{ }=

all positive integers

is a positive integer wh

30

: eere

is a positive integer where

x

B x x x

C

+ <{ }= − >{ }=

2 19

2 3 17:

xx x: is an odd integer<{ }30

(1) A BÈ(2) A BÇ

(3) AC

(4) B BCÈ

(5) A CCÇ

(6) C AC CÈ

(7) A B C∪( )∩(8) C B AC∪( )∩

(9) A BCC

∪( )(10) A UC È

B Solve the following problems using the Venn diagram and answer the related questions.

In a certain school, a group of students in a class were enrolled in three subjects.

Howmanystudentswereenrolledin:(11) exactly one subject?(12) exactly two subjects?(13) at most two subjects?(14) at most one subject?(15) Algebra or Geometry?(16) Algebra and Geometry?(17) Algebra and Geometry but not Trigonometry?(18) Geometry and Trigonometry but not Algebra?(19) Algebra only?(20) If there are 50 students in all, how many did not enroll in any of the three subjects?

U Algebra Geometry

Trigonometry

10

9 7

7

24

6

5

24

LINKING TOGETHER

Basic Idea of Sets

Describing Sets

Operations on Sets

Union of Set Intersection of Sets

Kinds of Sets

Rule Method Roster or Listing Method

Empty Sets Finite Sets InfiniteSets

Number of Subsets in a Set =2n

Disjoint Sets •Nocommon

elements

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