CHAPTER 11: SETS Definition of Sets: A sets is a collection of things or items which are well-defined and have common properties or features. Set Notation: Item Universal Set Subset
Symbols
ξ
A⊂ B
φ or {
Empty Set Equal Sets Elements of a set Complement of a set Intersection of two sets Union of two sets Number of elements in a set
}
A=B x ∈A A’ A∩ B A∪ B n(A)
Example The universal set ξ Set A is a subset of Set B The empty set φ or
{ }
Set A is equal to Set B x is an element in Set A The complement of set A The intersection of Set A and Set B The union of Set A and Set B The number of elements in Set A
Skills assessed • List the elements in a set that fulfills a certain description. • Use the Venn diagram to illustrate sets. • Use the concept of subset, universal set and the complement of a set. • Perform operations involving two or more sets: intersection and union. • List the elements for A ∩ B, A ∪ B, P ∩ Q’, P’ ∩ Q etc. • Determine the outcome of combined operations on sets and represent it in a Venn diagram. • Solve problems involving combined operations on sets. Solving strategy • List down all the elements in the universal set, ξ. • Mark or circle elements in different sets P, Q and R by using a different colour for each set. • To list the elements in a set, just identify the elements with the same colour. • For questions involving n(P ∩ Q), count the elements which belongs to both Set P and Set Q. • For questions involving n(P ∪ Q), count the elements which belongs to either Set P or Set Q. Common Mistakes • • of elements in the set. •
Students do not list down all the elements Students find the elements in the set while the question asked for the number 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 or empty set has no elements in it. The symbol for a null set is φ or
{}
=> If two sets, P and Q, contain exactly the same elements, then set P is equal to set Q, P = Q Sets
1
EXAMPLES
EXERCISES
1
List the elements and state the number of elements A = {factors of 18} Solution A = {1, 2, 3, 6, 9, 18} ∴ n(A) = 6
List the elements and state the number of elements A = {factors of 20} Solution A= ∴ n(A) =
2
List the elements and state the number of elements P = {consonants in the word ‘STUDENTS’} Solution P = {S, T, D, N, T} ∴ n(P) = 5
List the elements and state the number of elements P = {vowels in the word ‘UNIVERSAL’} Solution P= ∴ n(P) =
3
List the elements and state the number of elements P = {prime factors of 14} Solution P = { 2, 7}
List the elements and state the number of elements P = { prime factors of 21} Solution P= ∴ n(P) =
4
List the elements and state the number of elements P = {x : x is an integer, -2 ≤ x ≤ 4 } Solution P = {-2, -1, 0, 1, 2, 3, 4} ∴ n(P) = 7
List the elements and state the number of elements P = {x : x is an integer, -3 ≤ x ≤3 } Solution P= ∴ n(P) =
5
Given the sets P ={1, 2, 4, r, 8} and Q = {20, 21, 22, 23}. If P = Q, find the value of r. Solution P = {1, 2, r, 8} Q = {1, 2, 4, 8} ∴r=4
Given the sets R ={2, 4, m, 8} and T = {1 × 2, 2 × 2, 3 × 2, 4 × 2}. If R= T, find the value of m. Solution R= T= ∴m=
∴ n(P) = 2
Sets
2
11.2 (a)
SUBSETS.
Definition: Symbol:
If each element of set A is a member of set B, then set A is a subset of set B. A⊂ B.
Venn diagram
1
E n er ta in EXAMPLESm e nt
A⊂ B EXERCISE Given that X = { 4, 5} Y = {3, 4, 5, 6, 7} What is the relation between set X and Y?
Given that A = {p, q} B = {m, n, o, p, q} What is the relation between set A and B?
A⊂ B
Answer:
Answer: Each element in set _____ is a member of Set _________ .
Each element in set A is a member of set B. Venn diagram.
A
B
•p
2
•n
•q
•o
Draw the Venn Diagram.
•m
From the Venn Diagram below, what is the relation between set P,Q,R and S?
From the Venn Diagram below, what is the relation between set P, Q,R and S?
P Q
R
P S
S R
Answer:
Answer:
R ⊂Q Q ⊂P
S⊂P
R⊂P
Sets
3
Q
11.2 (b) UNIVERSAL SET Definition: a set which includes all the sets in a discussion. Symbol : ξ 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’ A’ Venn Diagram . ξ
A
. 1
EXAMPLES Given that ξ = {11,12,…20} A= {prime numbers} List all elements in set ξ, A and set A’. Then draw a Venn diagram for the sets. Answer : ξ = {11,12,13,14,15,16,17,18,19,20} A = {11,13,17,19} A’= {12,14, 15, 16, 18 ,20} ξ
ξ
A •11 •13 •17 •19
2 ξ
•12 •14 •15 •16 •18 •20 Q
•a •b •d
P
•c
ξ
•k •j •e •g
A
B
•5
•10 •6 •11 •7 •12
R
•f
•h •i
List the elements of the following sets ξ= {a, b, c, d, e, f, g, h, i, j, k} n(ξ)= 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
Sets
EXERCISE Given that ξ = {11,12,…20} B = {odd numbers} List all elements in set ξ, B and B’. Then draw a Venn diagram for the sets. Answer : ξ= B= B’=
•3 •1 •2 •4
C
•8
•9
List the elements of the following sets ξ= n(ξ)= A= n(A)= A’= n(A’)= B= n(B)= B’= n(B’)= C= n(C)= C’= n(C’)=
4
11.3
OPERATION ON SETS.
11.3 (a) INTERSECTION OF SETS. Definition : Symbol
The intersection of sets A and B is the set of elements which are common to both A and B.
: A∩B
The following Venn diagrams show three examples of the intersection of sets.
A
1
B
A
•y •z
2.
B
EXCERCISES It is given that A= {1, 2, 3, 4, 5} B = {4, 5, 6, 7} A∩B= Complete the Venn diagram below.
A
B
•w •x
B
•u •v
It is given that A= {p, q, r, s, t} B = {r, s ,t ,u ,v} C = {p ,s, t, u ,x}
It is given that P= {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 sets given. A B
Draw a Venn diagram to represent the sets given.
•q
•r •s •p •t •x
Sets
B
EXAMPLES It is given that A= {w, x, y, z} B = {u, v, w, x} A ∩ B = { w, x} Draw a Venn diagram to represent the sets given.
A
A
P
Q
•v •u C
R
5
3
EXAMPLES Shade the region that represents X ∩ Y ∩ Z for the following Venn diagrams X
i)
Y
EXCERCISES Shade the region that represents X ∩ Y ∩ Z for the following Venn diagrams i)
X
Y
Z
Z
ii) Z
Y
Y
X
Z
X
ii)
X
iii)
iii)
Z
X
Y
Y
Z
4
Find the complement of ( M ∩ N )' for each of the following sets. ξ={a, e, i, o, u} M={a, e, o, u} N= {i, o, u} ( M ∩ N ) = {o, u} ( M ∩ N )' = {a, e, i}
5.
a) ξ={2, 4, 6, 8, 10,12} M={4, 8, 12} N= {4, 6, 12} (M ∩ N ) = ( M ∩ N )' =
Shade the region that represents the complement of the following sets. a)
A
Find the complement of ( M ∩ N )' for each of the following sets.
Shade the region that represents the complement of the following sets. a)
A
B
B
()’
()’
b)
A
b)
A
B B
()’
()’ A
c)
B
A
( A ∩ B ∩ C )' C
11.3(b) Sets
B
UNION OF SETS.
6
( A ∩ B ∩ C )'
c) C
Definition: The union of sets A and B is the set of all elements which belong to A or B or both A and B. Symbol : A ∪ B
A
The following Venn diagrams show three examples of the union of sets.
B
A
A
B B
A∪ B
A∪ B
=A
EXAMPLES 1
EXCERCISE
Given that A= {w, x, y, z} B = {u, v, w, x}
Given that A= {1, 2, 3, 4, 5} B = {4, 5, 6, 7}
A ∪ B = {u, v w, x, y ,z} = {u, v w, x, y ,z} A∪ B
A
B
A •y •z
A∪ B = B
•u •v
•w •x
Complete the Venn diagram above. 2
Shade the region that represents X ∪ Y ∪ Z for the following Venn diagrams
Shade the region that represents X ∪ Y ∪ Z for the following Venn diagrams i)
i)
ii) ii)
iii) iii)
X
Y
X
Y
Z Z Sets
Z XX
Z Y Y
Y
X
Z
X
7
Y Z
EXAMPLES
EXCERCISES
3
It is given that It is given that ξ={2,4,6,8,10,12}
ξ ={a, e, i, o, u}
M={4,8,12}
M={a, e}
N= {4, 6}
N= { u} Find i) ( M ∪ N ) = {a, e,, u}
Find i)
ii) ( M ∪ N )' = {i, o}
(M ∪ N ) =
ii) ( M ∪ N )' =
4
Shade the region that represents the complement of the following sets.
Shade the region that represents the complement of the following sets.
a) a)
b) b)
c)
c)
B
A
()’
B
A
()’
A
A ()’
BB
Sets
B
B
A
( A ∪ B ∪ C )' 8
C
()’
A
B
( A ∪ B ∪ C )' C
11.3(c)
Sets
COMBINED OPERATIONS ON SETS.
9
EXAMPLES
EXCERCISES
1 ξ
ξ
Q
P
•a •c •b •d
Q
•k •j •e •g
•h •i
List the elements of a set
Sets
R P
•8
•9
List the elements of a set
ξ = {a, b, c, d, e, f, g, h, i, j, k} n(ξ)= 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( Q ∩ R)= 1
P ∩ R={}
n( P ∩ R)= 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
(Q ∪ P)’={f,g,h,i}
n[(Q ∪ P)’]=4
P’ ∩ Q’={f,g,h,i}
n( P’ ∩ Q’)=4
(P ∪ Q ∪ R)’={h,i}
•3 •1 •2 •4
•10 •6 •11 •7 •12
R
•f
•5
ξ=
n(ξ)=
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)=
P ∩ R=
n( P ∩ R)=
P’ ∩ R=
n( P’ ∩ R)=
Q’ ∩ R=
n( Q’ ∩ R)=
R’ ∩ Q=
n( R’ ∩ Q)=
Q ∪ R=
n( Q ∪ R)=
Q ∪ P=
n [(P ∪ Q ∪ R)’]=2
10
n( Q ∪ P)=
(Q ∪ R)’=
n[(Q ∪ R)’]=
(Q ∪ P)’=
n[(Q ∪ P)’]=
P’ ∩ Q’=
n( P’ ∩ Q’)=
(P ∪ Q ∪ R)’=
n [(P ∪ Q ∪ R)’]=
2
EXAMPLES Given that ξ = X ∪ Y ∪ Z Shade the region for the operation on the following sets.
EXCERCISES Given that ξ = X ∪ Y ∪ Z Shade the region for the operation on the following sets.
a) (X ∪ Y)
a) (Y ∪ Z)
∩Z
∩X
Y
X
X
Z
Z
b) (X ∩ Y)
Y
∪Z
b) (Z ∩ Y) Y
X
∪X X
Z
Y
Z
c) (X ∩ Y)’
∩
c) (Z ∩ Y)’
Z
∩
X X
Z X
Y Y
d)
X’ ∩ Y ∩ Z’ X
e)
Z
Y
d)
Z
X
X’ ∩ Y ∩ Z’
e)
Sets
Y
Z
X’ ∩ Y ∩ Z’
Z
Z X
Z’ ∩ Y ∩ X’
Y
Y
11
X
QUESTIONS BASED ON THE SPM FORMAT (Paper 1) 1.
EXAMPLES Given that the universal set ξ = P ∪ Q ∪ R , P ⊂ Q and Q ∩ R = φ , the Venn diagram that represents these relationships is
EXCERCISES Given that the universal set ξ = P ∪ Q ∪ R , R ⊂ Q and Q ∩ P = φ , the Venn diagram that represents these relationships is
A.
A.
ξ
R
P
ξ
Q
B.
B.
Q
P
ξ
R
P
C.
R
ξ
Q
Q
R
P
R
ξ
Q
R
P
Diagram shows a Venn diagram with the universal set ξ = P ∪ Q ∪ R .
P
Sets
D.
P
A
P
Q
ξ
2
Q R
ξ
D.
P
Q
ξ
C.
R
B
C
Q D
R
Diagram shows a Venn diagram with the universal set ξ = P ∪ Q ∪ R .
P
Q A
R B
C
D
Considering the four regions, A, B, C and D, which of these represents the set P ∩ R ∩ Q' ?
Considering the four regions, A, B, C and D, which of these represents the set P ∩ R ∩ Q' ?
Answer : B
Answer :
12
EXAMPLES 3
EXCERCISES
Diagram is a Venn diagram which shows the elements of sets P, Q and R.
Q
P •n
•q
•t
•r •s
Diagram is a Venn diagram which shows the universal set, ξ, and sets P, Q and R. ξ
R
•p •n
•q
•t
•r •s
If the universal set ξ = P ∪ Q ∪ R , then set R’ is A. {m ,n} B. {m, n, p, q} C. {t ,r, s} D. {m ,n , p , q ,t} Diagram is a Venn diagram which shows the universal set, ξ, and sets P, Q and R. ξ
The shaded region in the Venn diagram represents the set A. P ∩Q ' B P ∩ R' C. P '∩R D. R ∩Q '
P
The shaded region in the Venn diagram represents the set A. P ∩Q ' B P ∩ R' C. P '∩R R ∩Q ' D.
Q P
Q
R
R
Sets
Q
P
•m
If the universal set ξ = P ∪ Q ∪ R , then set Q’ is A. {r ,s} B. {n, t} C. {m, r, s} D. {n, p, q, t} 4
P
R
•p •m
Diagram is a Venn diagram which shows the elements of sets P, Q and R.
13
EXAMPLES 5
EXCERCISES
Given the universal set, ξ = {x : 10 ≤ x ≤ 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 ( P ∪ R ) ∩ Q' . A. {15, 18} B. {12, 18, 20} C. {12, 15, 16, 18, 20} D. {10, 12, 15, 16, 18, 20}
Given the universal set, ξ = {x : 5 ≤ x ≤ 25, x is an 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 ( P ∪ R ) ∩ Q' . 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. Q
ξ
P
3
Q
ξ
5 2 1
R
P
2
3 1
3
4
Diagram shows the number of elements in sets P, Q and R. Find n(Q ∪R )' A. 5 B. 6 C. 8 D. 12
R
2
Diagram shows the number of elements in sets P, Q and R. Find n(Q ∪R )' A 5 B 6 C. 8 D. 12
7. ξ
P
ξ
Q 2
5
R
1
4
3
3
2
5
4
3
Diagram shows the number of elements in sets P, Q and R. Find n( P ∪R )' . A. 4 B. 6 C. 8 D. 11
Sets
Q
R
1 3
P
14
Diagram shows the number of elements in sets P, Q and R. Find n( P ∪Q )' . A. 4 B. 7 C. 8 D. 11
EXAMPLES
EXCERCISES
8. ξ
ξ
P 2
1 Q
3 1
P
2 4
2
1 R
Q
3
2
x 1
A. B. C. D.
3
2
x
Diagram shows the number of elements in sets P, Q and R. Given that ξ = P ∪Q ∪ R. and n(P) = n(Q ∩ R) Find the value of x.
R
4
Diagram shows the number of elements in sets P, Q and R. Given that ξ = P ∪Q ∪ R. and n(R) = n(P ∩ Q) Find the value of x.
3 5 6 8
A. B. C. D.
3 5 7 9
9 ξ
ξ
P 2
y1 Q
3 x1
P
2 4
R
Q
43
5x
Sets
32 x1 3
2 54
R 3
4x
Diagram shows the number of elements in sets P, Q and R. Given that ξ = P ∪Q ∪ R. n(ξ)=30 and n(P’) = n(R). Find the value of x + y. A. B. C. D.
42
y1
5 7 8 10
Diagram shows the number of elements in sets P, Q and R. Given that ξ = P ∪Q ∪ R. n(ξ)=25 and n(P’) = n(R). Find the value of x + y. A. B. C. D.
15
5 7 8 10
10.
In the Venn diagram, given that
B
A
ξ = {x: x is an integer}
A = {x : x is a multiple of 5}
I
II I III
I
B = {x : x is a multiple of 7} Number 49 should be in the area A. B.
I II
C. D.
IV III IV
11. The Venn diagram shows the relationship between set M,N and K. Which of the following is true?. A. B. C. D.
ξ
M ∩ N=K M ∪ N=K M ∩ K=K K ∪ N=M
N
M K
12. Given ξ= { x : 1 ≤ x ≤ 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} E 13.
F G
Diagram shows a Venn diagram with the universal set ξ = E ∪ F ∪ G. Considering the four regions, A, B ,C and D, which of these represents the set E ∩ F ∩ G’ ?
A
B C
D
14. Diagram shows a Venn diagram with the universal set ξ= {Year Five pupils}, set M = { pupils who read a Malay language newspaper} and R= { pupils who read a English language newspaper}. Given that n(M) = 80, n (R) = 104, n(M ∩ R) = 12 and the number of pupils who do not read either newspaper is 4, find the total number of Year Five pupils.
Sets
A.
172
C.
196
B.
176
D.
200
16
ξ
M
R
15.
The diagram above shows a Venn diagram with universal set ξ, set P and set Q. The shaded region represents P ∩Q P ∩ Q’
A B 16.
C D
ξ
P Q
P’ ∩ Q P’ ∩ Q’
Given universal set ξ = A ∪ B ∪ C, set A = {K,E, D, A, H}, set B = {P, E, R, L, I, S} and set C = {S, E, L, A, N, G, O, R}. Find n[(A ∩ C) ∪ B] A B
17.
6 7
8 9
.Diagram shows a Venn Diagram with universal the ξ set ξ = {Form 4 Ibnu Sina students }, set P = {students who pass the Mathematics Paper 1} and set Q = { students who pass the Mathematics Paper 2} Given n(P) = 30, n(Q) = 22, n(P ∩ Q) = 20 and the number of students who fail both papers is 8, find the total number of students in Form Four Ibnu Sina. A B
18.
C D
38 40
C D
Q P
41 44
. The Venn diagram in diagram shows all the elements in the sets L, M and N.
•8
•1
The
Sets
17
A
elements of the set M’ are { 1, 8 }
B
{ 3, 7 }
C
{ 1, 3, 7, 8 }
D
{ 1, 2, 3, 7, 8 }
19.
The Venn diagram in diagram shows the number of elements in sets J, K and L. The universal set ξ = J ∪ K ∪ L .
If n( J ∪ K ) = 20 find the value of y.
20.
A
4
B
9
C
11
D
16
In the Venn diagram , the universal set, ξ = {Form 4 students} and the sets A = {Students who like basket ball} B= {Students who like badminton}
Given that n( A) = 60 , n( B) = 84 , n( A ∩ B ) = 16 and 6 of them dislike playing any games. Find the total of form four students.
Sets
A
128
B
134
C
144
D
148
18
Sets
19
Past Year SPM Questions Paper 1 November 2003 1
Given that the universal set ξ =J ∪K ∪L , J ⊂K and K ∩L = φ , the Venn diagram that represents these relationships is A.
B. ξ
L
ξ
J
D. J
ξ
L
K
L
J
K
2.
J
L
K
C. ξ
K
Diagram 13 shows a Venn diagram with the universal set ξ = E ∪ F ∪G .
F G
E A
B
C
D
Diagram 13
Considering the four regions, A, B, C and D, which of these represents the set E ∩ F ∩ G' ? 3.
Diagram 14 shows a Venn diagram with the universal set ξ= { Year Five pupils}, set M = {Pupils who read a Malay language newspaper} and R = { Pupils who read an English language newspaper }.
ξ
M
R
Given that n(M) = 80, n(R) = 104, n(M ∩ R) = 12 and the number of pupils who do not read either newspaper is 4, find the total number of Year Five pupils. A 172 C 196 B 176 D 200
Diagram 14
Sets
20
July 2004 It is given that the universal set ξ = E ∪ F ∪ G , 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?
4.
E
F
A.
B.
G
B.
E
F
D.
G
5.
F
G
F G
It is given that the universal set ξ= { x : 1 ≤ x ≤ 12, 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 B. 3, 5, 7
C. D.
1,2, 9 1, 9.
November 2004 6.
Diagram 13 is a Venn diagram which shows the elements of set P, Q and R.
Q
P •5
•6
R •3
•1 •2
•4
If the universal set ξ = P ∪ Q ∪ R , then set Q’ is A {1, 2} C {1, 2, 5} B {3, 6} D {3, 4, 6}
Sets
21
Diagram 13
E
E
Diagram 14 is a Venn diagram which shows the universal ξ, set P and set Q.
7.
ξ
P
Q Diagram 14
The shaded region in the Venn diagram represents the set
A. B. C. D. 8.
P’ ∩ Q’ P’ ∩ Q P ∩ Q’ P∩ Q
It is given that the universal set ξ= { x : 1 ≤ x ≤ 12, 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 (P ∪ R)’ A. 5, 11 B. 1, 5, 11
∩ Q are
C. D.
2, 3, 11 2, 3, 9.
July 2005 9. Given that M = {4, 6}, list all the subsets of M. A. {4}, {6} B. ø, {4}, {6} C. {4}, {6}, {4, 6} D. ø, {4}, {6}, {4, 6} 10. Diagram 13 is a Venn diagram which shows the elements of universal set ξ, set P and set Q.
ξ
•7
Q P •5
•1 •3
•2 •4
•6
•8
Sets
22
Diagram 13
List all the elements of set P’. A. B. C. D. 11.
{5, 6} {7, 8} (1, 2, 3, 4} {5, 6, 7, 8}
It is given that P = {3, 4,5, 6, 7}, Q = {6, 8, 10, 14}, R = {10, 16, 20} and universal set ξ = P ∪Q ∪ R. . Which of these Venn diagrams shows the relation between set P, Q and R?
A.
B.
P
C.
R
Q
D.
Q
P
November 2005.
12. Which of the following Venn diagrams represents set R ⊂ S such that the universal set ξ = R ∪ S ? R
S
R
A.
C.
S
S
S
R
B.
Sets
R
Q
P
R
Q
P
D.
23
R
R
13. Diagram 13 is a Venn diagram with the universal set ξ = P ∪Q ∪ R .
R
P
Q A
C
B
Diagram 13
D
Which of the region, A, B, C or D, represents the set P'∩Q '∩R? ? 14. Diagram 14 is a Venn diagram which shows the elements of set R, set S and set T. R
S 3
x-4 8
1 Diagram 14
x-3
9
5
T
It is given that the universal set ξ = R ∪ S ∪ T , and n(S’) = n(S Find the value of x. A. 5 C. 10 B. 6 D. 11
∩
T).
July 2006
15. Diagram 11 is a Venn diagram showing the sets M, N and P such that the universal set ξ =M ∪N∪P M N P
Diagram 11 The shaded region represents the set A B C D
( M ∪N) ∩P ( M ∩N) ∪P ( M ∩ N ') ∪ P ( M '∪ N ) ∩ P
16. Diagram 12 is an incomplete Venn diagram showing the number of elements in sets P, Q Sets
24
and R. P
Q 6
4 3
1
7 2
R
DIAGRAM 12
It is given that the universal set, ξ = P ∪ Q ∪ R and n ( R ) = 11 . Find n ( P ∩ Q ) ∪ R . A B C D
5 6 12 17
November 2006
17. It is given that the universal set, ξ = { x :19 ≤ x ≤ 31, x is an integer } and set
R = { x : x is a number such that the sum of its two digits is an even number} . Find set R ' . A B C D
{ 20, 22, 24, 26, 28 } { 21, 23, 25, 27, 29 } { 19, 21, 23, 25, 27, 29 } { 21, 23, 25, 27, 29, 30 }
18. Diagram 13 is a Venn diagram showing the universal set ξ , set P and set Q.
ξ P
Q
Diagram 13 Sets
25
The region ( P ∩ Q ) ' is equivalent to the region A 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 diagram that represents part of the information in Table 2. Programmes watched News Sports News and Sports only News and Entertainment only News only Sports only
Number of viewers 55 61 12 5 25 28
TABLE 2
12
News
28 Sports
5
Entertainment DIAGRAM 14
Find the number of viewers who watched sports or news and also entertainment programmes. A B C D
Sets
26 39 47 51
26