chapter 11 i sets enhance

8/3/2019 Chapter 11 i Sets Enhance

1/33

Sets 1

CHAPTER11: SETS

Definition of Sets: A sets is a collection of things or items which are well-defined and havecommon properties or features.

Set Notation:

Skills assessed

y List the elements in a set that fulfills a certain description.y Use the Venn diagram to illustrate sets.y Use the concept of subset, universal set and the complement of a set.y Perform operations involving two or more sets: intersection and union.y List the elements forA B, A B, P Q, P Q etc.y Determine the outcome of combined operations on sets and represent it in a Venn diagram.y Solve problems involving combined operations on sets.

Solving strategy

y List down all the elements in the universal set,\ .

y Mark or circle elements in different sets P, Q and Rby using a different colour for each set.y To list the elements in a set, just identify the elements with the same colour.y For questions involving n(P Q), count the elements which belongs to both Set P and Set Q.y For questions involving n(P Q), count the elements which belongs to either Set P or Set Q.

Common Mistakes

y Students do not list down all the elementsy Students find the elements in the set while the question asked for the number of elements in the set.y Students fail to extract information from a problem and to solve it using sets.

11.1: UNDERSTAND THE CONCEPT OF SET.

=> The number of elements in set A is written as n(A)

=> A null orempty set has no elements in it. The symbol for a null set is a_orJ

=> If two sets, P and Q, contain exactly the same elements, then set P is equal to set Q, P Q!

Item Symbols Example

Universal Set \ The universal set \

Subset A B Set A is a subset of Set B

Empty Set J or_ a The empty set J or_ a

Equal Sets A = B Set A is equal to Set B

Elements of a set x A x is an element in Set A

Complement of a set A

The complement of set A

Intersection of two sets A B The intersection of Set A and Set B

Union of two sets A B The union of Set A and Set B

Number of elements in a set n(A) The number of elements in Set A


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Sets 2

EXAMPLES EXERCISES

1 List the elements and state the

number of elementsA = {factors of 18}Solution

A = {1, 2, 3, 6, 9, 18}

@ n(A) = 6

List the elements and state the

number of elementsA = {factors of20}Solution

A =

@ n(A) =

2 List the elements and state thenumber of elements

P = {consonants in the word STUDENTS}Solution

P = {S, T, D, N, T}@ n(P) = 5

List the elements and state thenumber of elements

P = {vowels in the word UNIVERSAL}Solution

P =@ n(P) =


number of elementsP = {prime factors of 14}Solution

P =_ a2,7 @ n(P) = 2


number of elementsP = { prime factors of21}Solution

P =

@ n(P) =


number of elements

P = {x : x is an integer, -2 4ee x }

Solution

P = {-2, -1, 0, 1, 2, 3, 4}@ n(P) = 7


number of elements

P = {x : x is an integer, -3 3ee x }

Solution

P =@ n(P) =

5 Given the sets P ={1, 2, 4, r, 8} and

Q = {20, 2

1, 22, 2

3}.IfP = Q, find the value ofr.Solution

P = {1, 2, r, 8}Q = {1, 2, 4, 8}@ r = 4

Given the sets R={2, 4, m, 8} and

T = {1v2, 2v 2, 3 v2, 4 v 2}.IfR= T, find the value of m.Solution

R=

T =@ m =


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Sets 3

11.2 (a) SUBSETS.

Definition: If each element of set A is a member of set B, then set A is a subset of set B.

Symbol: BA .

Venn diagram

EXAMPLES EXERCISE

1 Given that

A = {p, q}

B = {m, n, o, p, q}What is the relation between set A and B?

Answer: BA

Each element in set A is a member of set B.

Venn diagram.

Given that

X = { 4, 5}

Y = {3, 4, 5, 6, 7}What is the relation between set X and Y?

Answer:

Each element in set _____ is a member ofSet _________ .

Draw the Venn Diagram.

2 From the Venn Diagram below, what is the

relation between set P,Q,Rand S?

Answer:

QR PQ

PS

PR

From the Venn Diagram below, what is the

relation between set P, Q,Rand S?

Answer:

A

B

p qn

o m

P

Q RS

P

R

S

Q

A

BA

B


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Sets 4

11.2 (b)UNIVERSAL SET

Definition: a set which includes all the sets in a discussion.

Symbol :

COMPLEMENT OF A SET.

Definition: Complement of set A is a set in the universal set which contains all the elements not

found in A.Symbol : A

.

. EXAMPLES EXERCISE

1 Given that = {11,12,20}

A= {prime numbers}List all elements in set , A and set A. Then

draw a Venn diagram for the sets.Answer : = {11,12,13,14,15,16,17,18,19,20}

A = {11,13,17,19}

A= {12,14, 15, 16, 18 ,20}

Given that = {11,12,20}

B = {odd numbers}List all elements in set , B and B. Then draw

a Venn diagram for the sets.Answer : =

B =

B=

2

List the elements of the following sets= {a, b, c, d, e, f, g, h, i, j, k} n()= 11P ={j, k} n(P)= 2

P= {a, b, c, d, e, f, g, h, i} n(P)= 9

Q = {a, b, c, d, e} n(Q)= 5Q= {f, g, h, i, j, k} n(Q)= 6

R= { e, f, g} n(R)= 3

R= {a, b, c, d, h, i, j, k} n(R)= 8

List the elements of the following sets= n()=

A = n(A)=

A= n(A)=

B = n(B)=B= n(B)=

C = n(C)=

C= n(C)=

A

A

11 1317 19

12 14

15 16

18 20

A

a c

b

d

PQ

Re

f

g

k j

h

i

C

56

710

1112

8

9

1 23

4

BA

Venn Diagram


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Sets 5

11.3 OPERATION ON SETS.

11.3 (a) INTERSECTION OF SETS.

Definition : The intersection of sets A and B is the set of elements which are common to both

A and B.

Symbol : BA

The following Venn diagrams show three examples of the intersection of sets.

EXAMPLES EXCERCISES

1 It is given that

A= {w, x, y, z}B = {u, v, w, x}

BA = { w, x}Draw a Venn diagram to represent the sets

given.

It is given that

A= {1, 2, 3, 4, 5}B = {4, 5, 6, 7}

BA =Complete the Venn diagram below.

2. It is given thatA= {p, q, r, s, t}

B = {r, s ,t ,u ,v}

C = {p ,s, t, u ,x}

Draw a Venn diagram to represent the setsgiven.

It is given thatP= {a, b, c, d, e, f}

Q = {d, e ,f ,g ,h}

R= {a ,d, e, h , i}

Draw a Venn diagram to represent the setsgiven.

BA

BA

A B

A B J !

A

B

BBA !

yz

uv

wx

A BA B

C

q vst u

A B

r

x

p

R

P Q


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Sets 6

EXAMPLES EXCERCISES

3 Shade the region that represents ZYX for the following Venn diagrams

i)

ii)

iii)

Shade the region that represents ZYX for the following Venn diagrams

i)

ii)

iii)

4 Find the complement of )'( NM for each ofthe following sets.

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

M={a, e, o, u}N= {i, o, u}

)( NM = {o, u}

)'( NM = {a, e, i}

Find the complement of )'( NM for each ofthe following sets.

a) ={2, 4, 6, 8, 10,12}

M={4, 8, 12}N= {4, 6, 12}

)( NM =

)'( NM =

5. Shade the region that represents the

complement of the following sets.a)

b)

c)

Shade the region that represents the

complement of the following sets.a)

b)

c)

X Y

Z

Z

YX

ZY

X

X Y

Z

Z

Y

X

X

Z

Y

B

C

A

)'( CBA

( BA )

A B

( BA )

A

B

)'( CBA

A B

C

( BA )

A

B

( BA )

A B


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8/33

Sets 8

EXAMPLES EXCERCISES

3It is given that

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

M={a, e}

N= { u}Find

i) )( NM = {a, e,, u}

ii) )'( NM = {i, o}

It is given that

={2,4,6,8,10,12}

M={4,8,12}

N= {4, 6}

Find

i) )( NM =

ii) )'( NM =

4 Shade the region that represents thecomplement of the following sets.

a)

b)

c)

Shade the region that represents the complementof the following sets.

a)

b)

c)

( BA )

( BA )

)'( CBA

A B

C

A B

B

( BA )

ABB ( BA )

)'( CBA

B

C

A

A B


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Sets 9

11.3(c) COMBINED OPERATIONS ON SETS.

EXAMPLES EXCERCISES


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Sets 10

1

List the elements of a set

= {a, b, c, d, e, f, g, h, i, j, k} n()= 11

P ={j, k} n(P)= 2

P= {a, b, c, d, e, f, g, h, i} n(P)= 9

Q = {a, b, c, d, e} n(Q)= 5

Q= {f, g, h, i, j, k} n(Q)= 6

R= { e, f, g} n(R)= 3

R= {a, b, c, d, h, i, j, k} n(R)= 8

P Q={} n( P Q)= 0

Q R={e} n( QR)= 1

PR={} n( PR)= 0

P R={e,f,g} n( P R)= 3

Q R= {f,g} n( Q R)= 2

R Q={a,b,c,d} n( R Q)= 4

Q R={a,b,c,d,e,f,g} n( Q R)= 7

Q P={a,b,c,d,e,k,j} n( Q P)= 6

(Q R)={k,j,h,i} n[(Q R)]=4

(QP)={f,g,h,i} n[(QP)]=4

P Q={f,g,h,i} n( P Q)=4

(P Q R)={h,i} n [(P Q R)]=2

List the elements of a set

= n()=

P = n(P)=

P= n(P)=

Q = n(Q)=

Q= n(Q)=

R= n(R)=

R= n(R)=

P Q= n( P Q)=

Q R= n( Q R)=

PR= n( PR)=

P R= n( P R)=

Q R= n( Q R)=

R Q= n( R Q)=

Q R= n( Q R)=

Q P= n( Q P)=

(Q R)= n[(Q R)]=

(Q P)= n[(QP)]=

P Q= n( P Q)=

(P Q R)= n [(P Q R)]=

EXAMPLES EXCERCISES

R

a cb

de

fg

k j

hi

PQ

P

56

710

1112

8

9

1 23

4

Q R


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Sets 11

2 Given that = XYZ Shade the region

for the operation on the following sets.

a) (XY) Z

b) (X Y) Z

c) (XY) Z

d)X

Y

Z

e) X Y Z

Given that = XYZ Shade the region

for the operation on the following sets.

a) (YZ) X

b) (Z Y) X

c) (ZY) X

d) Z

YX

e) XY Z

Z X

Y

Z X

Y

Z X

Y

X Y Z

Z

XY

X Y Z

XY

Z

XY

Z

X Y

Z

Z

YX


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Sets 12

QUESTIONS BASED ON THE SPM FORMAT (Paper 1)

EXAMPLES EXCERCISES

1. Given that the universal set

RQP !\ , QP and J! RQ ,the Venn diagram that represents theserelationships is

A.

B.

C.

D.

Given that the universal set RQP !\ ,

QR and J! PQ , the Venn diagram thatrepresents these relationships is

A.

B.

C.

D.

2 Diagram shows a Venn diagram with the

universal set RQP !\ .

Considering the four regions, A, B, C andD,

which of these represents the set

Q'RP ?

Answer : B

Diagram shows a Venn diagram with the

universal set RQP !\ .

Considering the four regions, A, B, C andD,

which of these represents the set Q'RP ?

Answer :

P

R

Q

P

Q

R

R

P

Q

R

Q

P

P

R

Q

P

Q

R

R

P

Q

R

Q

P

PRQ

CA B D

PR

Q

CA B D


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Sets 13

EXAMPLES EXCERCISES

3 Diagram is a Venn diagram which shows the

elements of sets P, Q and R.

If the universal set RQP !\ , then setQ is

A. {r ,s}B. {n, t}C. {m, r, s}D. {n, p, q, t}

Diagram is a Venn diagram which shows the

elements of sets P, Q and R.

If the universal set RQP !\ , then setR is

A. {m ,n}B. {m, n, p, q}C. {t ,r, s}D. {m ,n , p , q ,t}

4 Diagram is a Venn diagram which shows the

universal set, , and sets P, Q and R.

The shaded region in the Venn diagramrepresents the set

A. 'QP

B 'RP C. RP' D. 'QR

Diagram is a Venn diagram which shows the

universal set, , and sets P, Q and R.

The shaded region in the Venn diagramrepresents the set

A. 'QP

B 'RP C. RP' D. 'QR

P

P

QR

m n q

t r

s

pP

QR

m n q

t r

s

p

R

QP

R

QP


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Sets 14

EXAMPLES EXCERCISES

5 Given the universal set, = {x : 10 e x e 20,x is an integer},

P = {16,18,20}

Q = { x : x is prime number} and

R= {x : x is a multiple of 3}The elements of the set ')( QRP .

A. {15, 18}B. {12, 18, 20}C. {12, 15, 16, 18, 20}D. {10, 12, 15, 16, 18, 20}

Given the universal set, = {x : 5 e x e 25, x isan integer},

P = {12,16,18,20,25}

Q = { x : x is prime number} and

R= {x : x is a multiple of 5}The elements of the set ')( QRP .

A. {15, 18, 19}

B. {12, 18, 20, 25}C. {12, 15, 16, 18, 20, 25}D. {10, 12, 15, 16, 18, 20, 25}

6.

Diagram shows the number of elements in

sets P, Q and R. Find )'( RQn

A. 5B. 6C. 8D. 12

Diagram shows the number of elements in sets

P, Q and R. Find )'( RQn

A 5B 6

C. 8D. 12

7.


sets P, Q and R. Find )'( RPn .

A. 4B. 6C. 8D. 11

Diagram shows the number of elements in sets

P, Q and R. Find )'( QPn .

A. 4B. 7C. 8D. 11

PQ

R

3

21

5

3

P Q

R

2

1

4

3

2

P

Q

R

3 2

1

5 4

3

P

Q

R

3 2

1

5 4

3


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Sets 15

EXAMPLES EXCERCISES

8.


sets P, Q and R. Given that = .RQP

and n(P) = n(Q R) Find the value of x.

A. 3B. 5C. 6D. 8


sets P, Q and R. Given that = .RQP

and n(R) = n(P Q) Find the value of x.

A. 3B. 5C. 7D. 9

9

Diagram shows the number of elements insets P, Q and R. Given that = .RQP

n()=30 and n(P) = n(R). Find the value of

x + y.

A. 5B. 7C. 8D. 10

Diagram shows the number of elements insets P, Q and R. Given that = .RQP

n()=25 and n(P) = n(R). Find the value of

x + y.

A. 5B. 7C. 8D. 10

P

Q R

31

1

4

2

x2

2

P

Q R

3

x

1

1

4

2

32

P

Q R

3

x

1

1

4

2

32

P

Q R

3

x

1

1

4

2

32

P

Q R

4

5

x

y

4

2

32

P

Q R

3

x

1

1

4

2

32

P

Q R

3

4

x

y

5

4

2

2


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Sets 16

10. In the Venn diagram, given that

\ = {x:x is an integer}

A = {x : x is a multiple of 5}

B = {x : x is a multiple of 7}

Number 49 should be in the area

A. I C. IIIB. II D. IV

11. The Venn diagram shows the relationship

between set M,N and K. Which of the

following is true?.

A. M N = K

B. M N = KC. M K = KD. K N = M

12. Given \ = { x : 1 e x e 30 and x is an integer}P = { x : x is multiple of 6}

and Q = { x : x is number which contain the digit 2}.

Find P Q.A. {12, 18, 24}B. {12, 24}C. {12, 18, 24,32}D. {2, 12, 24}

13. Diagram shows a Venn diagram

with the universal set \ = E F G.Considering the four regions, A, B ,C andD,which of these represents the set

E F G ?

14. Diagram shows a Venn diagram with the universal set \ = {YearFive pupils}, set M= { pupils who read a Malay language newspaper} and R= { pupils who read a Englishlanguage newspaper}. Given that n(M) = 80, n (R) = 104, n(M R) = 12 and thenumber of pupils who do not read either newspaper is 4, find the total number of YearFive

pupils.

A. 172 C. 196

B. 176 D. 200

I III

AB

\

IV

II

E

F

G

A BC

D

\ M R

M

K

N\


17/33


18/33

Sets 18

19. The Venn diagram in diagram shows the number of elements in sets J, K and L.

The universal set LKJ !\ .

If 20)( ! KJn find the value ofy.

20. In the Venn diagram , the universal set,

\ = {Form 4 students} and the sets

A = {Students who like basket ball}

B= {Students who like badminton}

Given that 60)( !An , 84)( !Bn , 16)( ! BAn and 6 of them dislike playing any

games. Find the total of form four students.

A 128

B 134

C 144

D 148

A 4

B 9

C 11

D 16


19/33

Sets 19

Past Year SPM Questions

Paper 1

November 2003

1 Given that the universal set LKJ !\ , KJ and J!LK , the Venndiagram that represents these relationships is

A. B.

D.

C.

2. Diagram 13 shows a Venn diagram with the universal set GFE !\ .

Considering the four regions, A, B, C andD, which of these represents the set

'E GF ?

3. Diagram 14 shows a Venn diagram with the universal set \ = { YearFive pupils},

setM = {Pupils who read a Malay language newspaper} and R= { Pupils who read

an English language newspaper }.

\

J

L

K

J

K

L

L

J

K

L

K

J

E G

F

CA B D

MR

Given that n(M) = 80, n(R) = 104,

n(M R) = 12 and the number of pupilswho do not read either newspaper is 4, findthe total number of YearFive pupils.

A 172 C 196

B 176 D 200

Diagram 13

Diagram 14


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Sets 20

July 2004

4. It is given that the universal set GFE !\ ,set E = {2, 4, 6, 8, 10, 14, 16},set F = {2, 4, 6, 8, 10, 12}, and set G ={2, 4, 14},

Which Venn diagram shows the relation between set E, F and G?

A. B.

B. D.

5. It is given that the universal set \ = { ,121: ee xx x is an integer},

set P ={x ; x is an odd number} and set Q={x ; x is a prime number}.

The elements of the set P Q are

A. 1, 3, 5, 7 C. 1,2, 9B. 3, 5, 7 D. 1, 9.

Novemb

er2004

6. Diagram 13 is a Venn diagram which shows the elements of set P, Q and R.

If the universal set RQP!

\ , then set Q is A {1, 2} C {1, 2, 5}B {3, 6} D {3, 4, 6}

FG

E FG

E

F

G

EF

G

E

P

5y 6y

4y 3y

2

1

y

y

Q R

Diagram 13


21/33

Sets 21

7. Diagram 14 is a Venn diagram which shows the universal \ , set P and set Q.

\

The shaded region in the Venn diagram represents the set

A. P QB. P QC. P QD. P Q

8. It is given that the universal set \ = { ,121: ee xx x is an integer},

Set P = {2 ,3 ,7 ,9}, set Q = {x ; x is a prime number} and

Set R= {x : x is multiple of 4},

The elements of the set (PR) Q are

A. 5, 11 C. 2, 3, 11B. 1, 5, 11 D. 2, 3, 9.

July 2005

9. Given that M = {4, 6}, list all the subsets ofM.

A. {4}, {6}B. , {4}, {6}

C. {4}, {6}, {4, 6}

D. , {4}, {6}, {4, 6}

10. Diagram 13 is a Venn diagram which shows the elements of universal set \ ,

set P and set Q.

\

P Q

Diagram 14

P

Q

8

6

7

1

2

3 4

5 Diagram 13


22/33

Sets 22

List all the elements of set P.

A. {5, 6}B. {7, 8}C. (1, 2, 3, 4}D. {5, 6, 7, 8}

11. It is given that P = {3, 4,5, 6, 7}, Q = {6, 8, 10, 14}, R= {10, 16, 20} and

universal set .RQP !\ .

Which of these Venn diagrams shows the relation between set P, Q and R?

A. B.

C. D.

November 2005.

12. Which of the following Venn diagrams represents set SR such that the

universal set ?SR !\

A. C.

B.D

.

R

S

R

R

R

S

S

S

P Q R PQ R

P QR

P QR


23/33

Sets 23

13. Diagram 13 is a Venn diagram with the universal set RQP !\ .

Which of the region, A, B, C orD, represents the set R?P' 'Q ?

14. Diagram 14 is a Venn diagram which shows the elements of set R, set S and set T.

It is given that the universal set TSR !\ , and n(S) = n(S T).

Find the value of x.

A. 5 C. 10

B. 6 D. 11

July 2006

15.Diagram 11 is a Venn diagram showing the sets M, N and P such that the universal set

M N P\ !

The shaded region represents the set

A M N P B M N P C 'M N P D 'M N P

P

R

Q

CA B DDiagram 13

3

8

5

9

1x - 4

x - 3 Diagram 14

R S

T

M

NP

Diagram 11


24/33

Sets 24

16. Diagram 12 is an incomplete Venn diagram showing the number of elements in sets P, Q

and R.

It is given that the universal set, P Q R\ ! and 11n R ! .

Find n P Q R - .

A 5B 6

C 12D 17

November 2006

17. It is given that the universal set, _ a:19 31, is an integer x x x\ ! e e and set_ a: is a number such that the sum of its two digits is an even numberR x x! . Find set 'R .

A { 20, 22, 24, 26, 28 }B { 21, 23, 25, 27, 29 }

C { 19, 21, 23, 25, 27, 29 }D { 21, 23, 25, 27, 29, 30 }

18.Diagram 13 is a Venn diagram showing the universal set ,\ setPand set Q.

\

PQ

R

3

4 6 7

21

DIAGRAM 12

QP


25/33

Sets 25

Diagram 13

The region 'P Q is equivalent to the regionA 'P Q B

'P Q

C ' 'P Q D ' 'P Q

19. Table 2 shows data obtained from a survey of 100 television viewers. Diagram 14 is a Venn diagramthat represents part of the information in Table 2.

Programmes watched Number ofviewers

News 55

Sports 61

News and Sports only 12

News and Entertainment only 5

News only 25

Sports only 28

Find the number of viewers who watched sports or news and also entertainment programmes.

A 26

B 39C 47D 51

June 2007

20. It is given that the universal set, \ = { x : 13x5 ee , x is an integer }and set

P = { x : x is a prime number }.

List all the elements of set 'P .

A { 6, 8, 9, 10, 12 }B { 5, 7, 9, 11, 13 }C { 6, 8, 10, 12 }D { 5, 7, 11, 13 }

SportsNews

Entertainment

12 28

5

DIAGRAM 14TABLE2


26/33

Sets 26

21.Diagram 15 is a Venn diagram showing set P and set S. It is given thatP = { hockey player }, S = { badminton layers } and the universal set , SP!\ .

If n ( P ) = 27, n ( S ) = 36 and )SP(nfind,8)SP( !n .

A 71

B 63C 55D 47

Nov 2007

22.List all the subsets of set X= { m , n }

A { m } , { n }B { m } , { n } , { }C { m } , { n } , { m , n }D { m } , { n } , { m , n } , { }

23.Diagram 14 is a Venn diagram showing the universal set , \ = { Form Five students }, set

L = { Students who play badminton } and set M = { Students who play tennis }

Given n (\ ) = 60, n (L ) = 25, n ( M ) = 12 and 8)PL( !n , find the number of

students who do not play the two games.

A 15B 23C 31

D 52

L M\

Diagram 14

P S

Diagram 15


27/33

Sets 27

24. Given the universal set, ,QP !\ set P = { 0 , 1, 2 } and set Q = { 2, 3, 4, 5 }, find the

value of n (\ ).

A 7B

6C 5D 4

Jun 2008

25.Diagram 12 shows a Venn diagram with the universal set, \ , set T and set S.

List all the elements of set S' .

A { 1, 4 }B { 3, 7 }C { 1, 4, 5 }D { 1, 3, 4, 7 }

26.Diagram 13 shows a Venn diagram with universal set , \ = { Form Five students }, set P =

{ Members of Science Society } , set Q = { Members ofMathematics Society } and set R=

{ Members of Geography Society }.

It is given that n (P ) = 82, n (Q ) = 90, n (R) = 40 and 24)QP( !n . The number of

Form Five students who are not a member of any of the three societies is 35.Find the total number ofForm Five students.

T S\

y 1

y 4y 7

y 6y 5 y 2

y 3

Diagram 12

Diagram 13

P Q\ R


28/33

Sets 28

A 271

B 236C 223D 188

27.Diagram 14 is a Venn diagram showing the number of elements in sets P , Q and R.

It is given that the universal set, RQP !\ . Find ))RQ(P( n

A 5

B 6C 10D 17

28.Diagram 13 is a Venn diagram showing the elements of the universal set, \ , set T and set U.

List all the elements of set U' .

A { q}

B { l, p }C { l, p , q }D { l, p , q , n}

R

P

Q

1

27 3

4

Diagram 14

T U\

y l yny my k

y q

yp

Diagram 13


29/33

Sets 29

Nov 2008

29.Diagram 14 shows a Venn diagram with the universal set , SR!\

Given that, n (R) = 95, n (S ) = 78 and 25)SR( !n , find n (\ ).

A 123B 148C 173D 198

30. Diagram 15 is a Venn diagram showing the number of students in sets G, M and S. It is

given that set G = { Geography Society Members } , set M = { Mathematics SocietyMembers }, set S = { Science SocietyMembers } and universal set, SMG !\ .

If the number of students who join both the Science Society and the Mathematics Society

is 8, find the number of students who join only two societies.

A 13B 15C 16D 18

R S

Diagram 14

Diagram 15

G

S

M

4

4

22x

x 7

6


30/33

Sets 30

ANSWERS

Chapter 11 Sets

3.1: UNDERSTANDTHECONCEPT OF SET.

1.A={1,2,4,5,10,20}

n(A) = 6

2.P={A,E,I,U}

n(P) = 4

3.P={3,7}

n(P) = 2

4.P={-3,-2,-1,0, 1,2,3}

n(P) = 7

5.m = 6

3.2 (a) SUBSET 3.2 (b) UNIVERSAL SET

1 X Y

All elemens in set X is a member of

set Y.

1 =

{11,12,13,14,15,16,17,18,19,20}B = {11,13,15,17,19}

B= {12,14,16, 18 ,20}

2 PR

SQ

PS

PQ

2 = {1,2,3,4,5,6,7,8,9,10,11,12} n()=12

A = {3,4,5,6,7} n(A)=

5

A= {1,2,8,9,10,11,12} n(A)=7

B = {1,2,3,4} n(B)=4B= {5,6,7,8,9,10,11,12} n(B)=

8

C = {8,9} n(C)=

2C= {1,2,3,4,5,6,7,10,11,12} n(C)=

10

Y7X

4 5

63

12 14

16

18 20

B

11 17

13

15 19


31/33

Sets 31

3.3.a INTERSECTION OF SETS. 3.3(b) UNION OF SETS.

1.BA = {4,5}

1BA = {1,2,3,4,5,6,7}

2 2 i

3 i)

ii)

iii)

ii)

iii)

4. )( NM = {4,12}

)'( NM = {2,6,8,10}

3 )( NM = {4,6,8,12}

)'( NM = {2,10}

5 a)

b)

c)

4a)

b)

c)

P

R

b

c

g

ed h

Q

f

i

a

Y

X

X

Z

Y

A

A B

Z

X Y

Z

Z

Y

X

X

Y

B

12

3

67

B

.4

.5

A

12

3

67

.4

.5


32/33

Sets 32

3.3(c) COMBINED OPERATIONS ON SETS.

1

List the element of set

= {1,2,3,4,5,6,7,8,9,10,11,12} n()= 12

P ={8,9} n(P)=2

P= {1,2,3,4,5,6,7,10,11,12} n(P)=10

Q ={3,4,5,6,7} n(Q)=5

Q={1,2,8,9,10,11,12} n(Q)=7

R= {1,2,3,4} n(R)=4

R= {5,6,7,8,9,10,11,12} n(R)=8

P Q={} n( P Q)= 0

Q R= {3,4} n( Q R)=2

PR= {} n( PR)=0

P R={1,2,3,4} n( P R)=4

Q R= {1,2} n( Q R)=2

R Q= {5,6,7} n( R Q)=3

Q R={1,2,3,4,5,6,7} n( Q R)=7

Q P= {3,4,5,6,7,8,9} n( Q P)=7

(Q R)={8,9,10,11,12} n[(Q R)]=5

(Q P)= {1,2,10,11,12} n[(QP)]=5

P Q= {1,2,10,11,12} n( P Q)=5

(P Q R)= {10,11,12} n [(P Q R)]=3

2a)

b)

c)

d)

e)

P

56

710

1112

8

9

1 23

4

Q R

Y

Z

X

Z X

Y

Z X

Y

Z

X

Y

X Y Z


33/33

SPM FORMAT QUESTIONS

1. B 2. C 3. B 4. C 5. D

6. A 7. B 8. C 9. A 10 C11. C 12 B 13 C 14 B 15 C

16. B 17 B 18 C 19 B 20. B

SPM PAST YEARS QUESTIONS (SETS)

2003. 1. D 2. C 3. B

2004. 4. C 5. C 6. A

2004J 7. B 8. D

2005 9. D 10. C 11. C

2005J 12 D 13. D 14 B

2006J 15 B 16 D

2006 17 D 18 C 19 A

2007J

20A

21C

22

D

2007 22 D 23 C 24 B

2008J 25 D 26 D 27 B

2008 28 C 29 B 30 C

chapter 11 i sets enhance

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