Chapter 5 II Set Enhance

8/3/2019 Chapter 5 II Set Enhance

1/14

Sets 1

Chapter5: SETS

Usual format of questions:

1. Giventhe shadedregionofa Venndiagram, statetheoperations thatareinvolved.

2. Givena Venndiagram, shadetheregion whichrepresents acombinationofsets basedontheoperationofsets.

Examples of Venn diagrams

A

A

AB

A B =

AB

AB

A

B

BA

A

B

AB

A

B

ABBA ,

A

B

J{BA

PQ

PQ P Q

P Q

R

'QP

')( QP

RQP

P Q

R

'

RQP

P Q

R

'

)(R

QP

PQ

PQ

R

R

R

PRQR , PRQR , J! RPQR ,


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Sets 2

Understanding Concepts concerning Sets

EXAMPLES EXERCISES

1 Given set A is thefactorof18.Then set A is

Answer18 = 1 X 18

= 2 X 9= 3 X 6

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

Given setGis thefactorof36.Then setGis

2

Giventhat P = {primenumbers thatareless

than 20 },findn(P).

Answeri. Whatareprimenumbers?

ii. Listtheelements in P.

P = { 2, 3, 5, 7, 11, 13, 17, 19 }

iii. Thencountalltheelements listed,n(P) = 8

Given Y = {primefactorof10},findn(Y).

3. Giventhat = { x : 1 < x 16 , x areintegers},A = { x : x areperfect squares },B = { x : x are multiples of3}

andC = { x : x areprimenumbers}.

a) Listoutallelements in set A.b) Findn(A B C).

c) Listoutalltheelements in

(A B C)

Steps :1. Define , A, B andC.

= { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,

13, 14, 15, 16 }

A = { 4, 9, 16}B = { 3, 6, 9, 12, 15 }

C = { 2, 3, 5, 7, 11, 13)

2. Constructthe Venndiagram.

Answera) {4, 9, 16}b) 0c) {8, 10, 14}

Giventhat = { x : x areintegers suchthat5 x 16 },

set P = { x : x areevennumbers },

set Q = { x : x aremultiples of6 },

set R = { x : x areperfect squares}.

a) Constructthe Venndiagramto showthe

relationbetween set , P, Q and R.

b) Findn(P Q R).c) Listoutalltheelements in

(P Q R)


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Sets 3

EXAMPLES EXERCISES

4. The Venndiagram shows J, Kand L sets with

= J K L. Shadetheregionforthe

followingoperations(a) J (K L) ondiagram (i)

(b) ( J K) L ondiagram (ii)

Answer:

(i) (ii)

(i) - shadethe J set- shadethe (K L) region

- thencombinethe shadedregion

(ii)- Shade set K, combine with set Jbutnotincluding L set

The Venndiagram shows A, B andC sets with

= A B C. Shadethe regionforthe

followingoperations(a) A B C ondiagram (i)

(b) A ( B C) ondiagram (ii)

(i) (ii)

5. Shadetheregionforthefollowingoperationsa) (P Q) R

b) (P R) Q

Answer:

a) (P

Q) shade Q butnotinclud

ing P,then shade R.

b) (P R) shadeallexcept P andR,then shade Q. Theintersectionis thecorrectarea.

Onthefollowing Venndiagrams, shadetheareaof

a) P Q R

b) (P Q) R

P Q

R

P Q

R

B C B C

A A

P QR

P Q R

J

L

J

K KL


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Sets 4

EXAMPLES EXERCISES

6.

If = { x : x areintegers where 12 x < 30}A = {15, 20, 13, 14 }

B = {14, 16, 18, 20, 22}C = {18, 20, 24, 28, 29}.

Find

a) A B Cb) n (A B C)

c) (A B)d) (A B) C.

Answer:

1. Constructthe Venndiagram

a) {13, 14, 15, 16, 18, 20, 22, 24, 28, 29}

b) 10 (count)

c) {12, 17, 19, 21, 23, 24, 25, 26, 27, 28,

29}

d) {18, 24, 28, 29}

Giventhat = { x : x areintegers where2 x 16}

set P = { 2, 4, 6 8, 10, 12, 14}set Q = { 3, 6, 9, 12, 15},andset R = { 5,10,12,15}.

Finda) P Q R

b) n (P Q R)c) (P Q)

d) (P Q) R.

Answer:

.20A

B

C

.14

.18

.15

.13

.16

.22

.24 .28

.29

.12

.17

.19

.21

.23

.25

.26

.27


5/14

Sets 5

Questions based on the Examination Format (Paper 2).

1. The Venndiagramintheanswerspace shows sets A, B andC. Itis giventhat

= ABC.

Onthediagramprovidedintheanswerspaces, shadetheareaof,

(a) C (AB)

(b) (A C)(B C)

Answer:

2.

9

8-x

7-x

89-x7

Giventhat CBA !\ andthat 34)( !\n ,find

(a) thevalueofx

(b) )'( CBAn

A B

x

C


6/14

Sets 6

3. Giventhattheuniversal set, RQP !\ .

The Venndiagrams below show sets P, Q and R. Onthediagrams, shade

(a) QP (b) 'RP

P

Q

R

P

Q

R

(c) '' RQP

PQ

R

4. Giventheuniversal set, xxx ,2510:{ ee!\ is aninteger}

Set P = {x : x is aprimenumber}Set Q = {x : x is amultipleof4} and

Set R = {x: x is anumberwhereoneofits digits is morethan 7}

(a) Findtheelements ofset P

(b) Findtheelements ofset P R.(c) Find n( RQP ).

5. The Venndiagram shows the set E, F andG. Iftheuniversal setis GFE !\ ,

intheanswerpart, shadetheregions for(a) FE

(b) GFE

Answer:

(a) E F G (b) E F G

6. Given P = {3, 5}


7/14

Sets 7

Q = {2, 4, 7, 8}

R = {1, 3, 5, 7, 9}anduniversal set RQP !\

(a) Draw a Venndiagramto show therelationshipbetween setS P, Q and R.

(b) Listalltheelements in set .RP

7. The Venndiagramintheanswerspace shows sets P, Q and R. Onthediagram

providedintheanswerspaces, shadea) the set Q R,

b) the set P Q R

Answer:a) b)

8. The Venndiagramintheanswerspace shows sets P, Q and R. Giventhat

universal set = P Q R. Intheanswerspaces, shadea) Q (P R),

b) (Q R) P.

Answer:

a)

9. Giventhatuniversal set = { x : 31 x 45, x areintegers}.Set P = { x : x areprimenumbers }

Set Q = { x : x areoddnumbers }Set R = { x : x is anumberwherethe sumofits digits is morethan 8 }

a) Listoutalltheelements in set R,

b) Findn(P Q R).

P

Q

R

Q

R

P

P R

P

Q RQ


8/14

Sets 8

10. The Venndiagramintheanswerspace shows sets P, Q and R. Giventhattheuniversalset = P Q R. Ontheanswerspaces, shade

a) (P Q) U R b) (P U R) Q

Past Year SPM Questions

Paper 2

1. June 2004

The Venndiagramintheanswerspace shows sets P, Q and R suchthattheuniversal set

= P Q R.Onthediagramintheanswerspace, shade

a) the set Q.b) the set P (Q R).

Answer:

a) b)

2. November 2004

The Venndiagramintheanswerspace shows sets A, B andC. Onthediagramprovideintheanswerspaces, shade

a) the set A Bb) the set A B C.

P R P

Q

R

QP R

QP R


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Sets 9

Answer:

a) b)

3. June 2005

The Venndiagramintheanswerspace shows sets P, Q and R suchthattheuniversal set = P Q R.

Onthediagramintheanswerspace, shadea) the set P R.

b) the set (P Q) R.

Answer:a) b)

4. June 2006

The Venndiagramintheanswerspace shows sets P, Q and R suchthattheuniversal set = P Q R.Onthediagramintheanswerspace, shade

a. the set P R.b. the set (P R) Q.

Answer:

c) b) QP

R

QP

R

A B

C

A BC

QP

R

QP

R


10/14

Sets 10

5. November 2006

The Venndiagramintheanswerspace shows sets P, Q and R suchthattheuniversal set,

P Q R\ ! .

Onthediagrams intheanswerspace, shade

a) the set'P Q

b) the set 'P Q R

Answer:

a) b)

6. June 2007

The Venndiagrams intheanswerspace shows sets K, L and M suchthattheuniversalset, MLK !\ .

Onthediagrams intheanswerspace, shadethe set

a) MK' ,

b) )M'(LK . [3 marks]

Answer:a)

b)

P

QR

P

QR

L

K

M

L

K

M


11/14

Sets 11

7. Nov 2008

The Venndiagrams intheanswerspace shows sets P, Q and R suchthattheuniversal set,

RQP !\ .

Onthediagrams intheanswerspace, shade [3 marks]

a) QP ,

b) R)(P ' Q .

Answer

a)

b)

R

P Q

R

P Q


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Sets 12

ANSWERS


13/14

Sets 13


14/14

Sets 14

SPM PAST YEARQUESTION (PAPER 2)

1 2004J (a) (b)

2 2004N (a) (b)

3 2005J (a) (b)

4 2006J (a) (b)

5 2006N (a) (b)

6 2007J (a) (b)

7 2008N (a) (b)

QP

R

QP

PR

PR

QP

R

QP

R

A B

C

A B C

P

Q R

P

Q R

L

KM

L

KM

R

P Q

R

P Q

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Chapter 5 II Set Enhance