Chapter 5: SETS Usual format of questions: 1. Given the shaded region of a Venn diagram, state the operations that are involved. 2. Given a Venn diagram, shade the region which represents a combination of sets based on the operation of sets. Examples of Venn diagrams A
ξ
A
ξ
B
B
A
A’
B⊂A
B
B A
A ⊄ B, B ⊄ A
ξ
A∩ B ≠ φ
ξ
A∩B Q
P
ξ
P ∩Q
Q
ξ
( P ∩Q )
'
R
R
P ∩ (Q ∪ R ) '
P ∪Q ∩ R '
P ∩Q ∩ R
Q
Q
ξ
ξ
P
'
P
Q
P
R
ξ
Q
P
A∪B
P
A
A⊂B
ξ A ∩ B =∅
ξ
A
ξ
B
ξ
ξ
P
R
Q
ξ
P
Q
R
R ⊂ Q, R ⊄ P Sets
R ⊂ Q, R ⊂ P
1
R R ⊄ Q, P ∩ R = φ
Understanding Concepts concerning Sets EXAMPLES 1
EXERCISES
Given set A is the factor of 18. Then set A is
Given set G is the factor of 36. Then set G is
Answer 18 = 1 X 18 =2X9 =3X6 Therefore A = { 1, 2, 3, 6, 9, 18 } 2
Given that P = { prime numbers that are less than 20 }, find n(P).
Given Y = {prime factor of 10}, find n(Y).
Answer i. What are prime numbers? ii. List the elements in P. P = { 2, 3, 5, 7, 11, 13, 17, 19 } iii. Then count all the elements listed, n(P) = 8 3.
Given that ξ= { x : 1 < x ≤ 16 , x are integers}, A = { x : x are perfect squares }, B = { x : x are multiples of 3} and C = { x : x are prime numbers}. a) List out all elements in set A. b) Find n(A ∩ B ∩ C). c) List out all the elements in (A ∪ B ∪ C)’
a) Construct the Venn diagram to showthe relation between set ξ, P, Q and R. b) Find n(P ∩ Q ∩ R). c) List out all the elements in (P ∪ Q ∪ R)’
Steps : 1. Define ξ , A, B and C. ξ = { 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. Construct the Venn diagram. Answer a) {4, 9, 16} b) 0 c) {8, 10, 14}
Sets
Given that ξ = { x : x are integers such that 5 ≤ x ≤ 16 }, set P = { x : x are even numbers }, set Q = { x : x are multiples of 6 }, set R = { x : x are perfect squares}.
2
4.
EXAMPLES The Venn diagram shows J, K and L sets with ξ = J ∪ K ∪ L. Shade the region for the following operations (a) J ∪ (K ∩ L) on diagram (i) (b) ( J ∪ K) ∩ L’ on diagram (ii)
EXERCISES The Venn diagram shows A, B and C sets with ξ = A ∪ B ∪ C. Shade the region for the following operations (a) A ∩ B ∩ C’ on diagram (i) (b) A’ ∩ ( B ∪ C) on diagram (ii)
Answer: (i)
(ii)
J
K
(i)
J
L
K
(ii)
A
A
L B
(i) - shade the J set - shade the (K ∩ L) region - then combine the shaded region
C
B
C
(ii)- Shade set K, combine with set J but not including L set 5.
Shade the region for the following operations a) (P’ ∩ Q) ∪ R ξ
P
Q
On the following Venn diagrams, shade the area of a) P ∩ Q ∩ R
R
P b) (P
∪
ξ
P
R)’
Q
∩Q Q
R
R b) (P’
Answer:
P
∩
Q’)
∩R Q
a) (P’ ∩ Q) – shade Q but not including P, then shade R.
R
b) (P ∪ R)’ – shade all except P and R, then shade Q. The intersection is the correct area.
Sets
3
EXAMPLES
EXERCISES
6. If ξ = { x : x are integers where 12 ≤ x < 30} A = {15, 20, 13, 14 } B = {14, 16, 18, 20, 22} C = {18, 20, 24, 28, 29}. Find
A∪ B∪ C n (A ∪ B ∪ C) (A ∪ B)’ (A ∩ B)’ ∩ C.
a) b) c) d)
Find
a) b) c) d)
Answer: 1. Construct the Venn diagram ξ .23
A
.15
.26
.13
.14 .20 .24
.25 .27 .21
.12
.22
.17
.18 .29 .28
.19
C
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}
Sets
P∪ Q∪ R n (P ∪ Q ∪ R) (P ∪ Q)’ (P ∩ Q)’ ∩ R.
Answer:
B
.16
Given that ξ = { x : x are integers where 2 ≤ x ≤ 16} set P = { 2, 4, 6 8, 10, 12, 14} set Q = { 3, 6, 9, 12, 15}, and set R = { 5,10,12,15}.
4
Questions based on the Examination Format (Paper 2) .
1.
The Venn diagram in the answer space shows sets A, B and C. It is given that ξ = A ∪ B ∪ C. On the diagram provided in the answer spaces, shade the area of , (a) C′ ∩ (A ∪ B) (b) (A′ ∩ C) ∪ (B ∩ C) Answer :
2.
B
A 7
9-x
x
8-x
8 7-x
9
C
Given that ξ = A ∪ B ∪ C and that n(ξ) = 34 , find (a) (b)
Sets
the value of x
n( A ∩ B ∩ C ' )
5
3. Given that the universal set, ξ = P ∪ Q ∪ R . The Venn diagrams below show sets P, Q and R. On the diagrams, shade P ∩Q
(a)
(b) Q
P
P ∪ R' Q
P R
R
P '∩Q ∩R '
(c)
P
Q R
Given the universal set,
4.
ξ = {x :10 ≤ x ≤ 25, x
is an integer}
Set P = {x : x is a prime number} Set Q = {x : x is a multiple of 4} and Set R = {x: x is a number where one of its digits is more than 7} (a) (b) (c) 5.
Find the elements of set P Find the elements of set P ∩ R. Find n( P ∪Q ∪ R )’.
The Venn diagram shows the set E, F and G. If the universal set is ξ = E ∪ F ∪ G , in the answer part, shade the regions for (a) E ∪ F (b) ( E ∩ F ) ∪ G Answer : (a)
Sets
E
F
G
(b)
6
E
F
G
6.
Given
P = {3, 5} Q = {2, 4, 7, 8} R = {1, 3, 5, 7, 9} and universal set ξ = P ∪ Q ∪ R
(a) (b) 7.
Draw a Venn diagram to show the relationship between setS P, Q and R. List all the elements in set P ∪ R.
The Venn diagram in the answer space shows sets P, Q and R. On the diagram provided in the answer spaces, shade a) the set Q’ ∩ R, b) the set P ∪ Q ∩ R Answer: a)
P
8.
b) Q
P
Q
R
R
The Venn diagram in the answer space shows sets P, Q and R. Given that universal set ξ = P ∪ Q ∪ R. In the answer spaces, shade a) Q ∩ (P ∪ R), b) (Q ∪ R)’ ∩ P. Answer: a) P
Q
Q P R
R
9.
Given that universal set ξ = { x : 31 ≤x ≤ 45, x are integers}. Set P = { x : x are prime numbers } Set Q = { x : x are odd numbers } Set R = { x : x is a number where the sum of its digits is more than 8 } a) b)
Sets
List out all the elements in set R, Find n(P ∪ Q ∪ R)’.
7
10.
The Venn diagram in the answer space shows sets P, Q and R. Given that the universal set ξ = P ∪ Q ∪ R. On the answer spaces, shade
a)
(P ∩ Q) U R
b)
(P U R)’ ∩ Q
Q
Q
R
P
R
P
Past Year SPM Questions Paper 2
July 2004 1.
The Venn diagram in the answer space shows sets P, Q and R such that the universal set ξ = P ∪ Q ∪ R. On the diagram in the answer space, shade a) the set Q’. b) the set P ∩ (Q ∪ R). Answer: a) P
b)
R Q
P
R Q
November 2004
2.
Sets
The Venn diagram in the answer space shows sets A, B and C. On the diagram provided in the answer spaces, shade a) the set A ∩ B’ b) the set A ∪ B ∩ C’.
8
Answer: a)
B
A
A
B
b)
C
C
July 2005 3.
The Venn diagram in the answer space shows sets P, Q and R such that the universal set ξ = P ∪ Q ∪ R. On the diagram in the answer space, shade a) the set P ∩ R’. b) the set (P’ ∪ Q’) ∩ R. Answer: a)
P
b)
Q R
P
Q R
July 2006 4.
The Venn diagram in the answer space shows sets P, Q and R such that the universal set ξ = P ∪ Q ∪ R. On the diagram in the answer space, shade a. the set P ∪ R. b. the set (P ∩ R) ∪ Q’. Answer: c) b)
Sets
P
Q
P
R
9
Q R
November 2006 5. The Venn diagram in the answer space shows sets P, Q and R such that the universal set, ξ = P ∪Q ∪ R . On the diagrams in the answer space, shade a) the set P ' ∩ Q
b) the set ( P ∪ Q ') ∩ R Answer: a)
b)
Q
R
R P
P
Sets
Q
10