The document contains 3 examples of SPM (Malaysian public exam) questions on sets. The first question involves finding the complement of a set R defined in terms of digit sums. The second uses a Venn diagram to represent the complement of a union of sets P and Q. The third is a word problem using a table and Venn diagram about TV viewership data. The examples are intended to help students practice set theory concepts for the SPM exam.

1. NEXUS SPM MATHEMATICS FORM 4 & 5
Page 41 (More Cloned SPM Questions) Supplementary Materials for
CHAPTER 3: SETS
Cloned SPM Question (2006, Paper 1)
Given the universal set ξ = {x :11 ≤ x < 21, 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 R'.
A {11, 13, 15, 17, 19, 20} C {12, 14, 16, 18}
B {12, 14, 16, 18, 21} D {12, 14, 16}
Solution
ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
R = {11, 13 15, 17, 19, 20}
R' = {12, 14, 16, 18}
Answer: C
Pointers
• Be careful with the definition of the universal set, ξ. The integer 21 is not an element of ξ
because of the inequality < 21.
• R' is the complement of R. That is it contains elements which are not in R, but in ξ.
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2. NEXUS SPM MATHEMATICS FORM 4 & 5
Page 41 (More Cloned SPM Questions) Supplementary Materials for
Cloned SPM Question (2006, Paper 1)
The Venn diagram below shows the relationship between the universal set ξ, set P and set Q.
ξ
P Q
The region (P ∪ Q)' is equivalent to the region
A P' ∩ Q' C P ∩ Q'
B P' ∪ Q' D (P ∩ Q)'
Solution
(P ∪ Q)' can be represented by the shaded region as shown in the Venn diagram below.
ξ
P Q
The shaded region is equivalent to P' ∩ Q'.
Answer: A
Pointers
• (P ∪ Q)' means the complement of (P ∪ Q), that is the region outside (P ∪ Q) or the
shaded region as shown in the Venn diagram.
• Since the shaded region also represents the overlap of the regions outside P and outside Q,
the region is also equivalent to P' ∩ Q'.
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3. NEXUS SPM MATHEMATICS FORM 4 & 5
Page 41 (More Cloned SPM Questions) Supplementary Materials for
Cloned SPM Question (2006, Paper 1)
The table on the left below shows the data obtained from a survey of 120 television viewers. The
diagram on the right is a Venn diagram that represents part of the information shown in the table.
Number of
Programmes Watched News
Viewers
19
Entertainment 70
Sports 64 13
Sports and Entertainment only 16
News and Entertainment only 13
Sports 16
News only 19 Entertainment
Sports only 21
Find the number of viewers who watched sports or news and also entertainment programmes.
A 40 B 43 C 46 D 64
Solution
Let the number of viewers who watched sports and entertainment and also news programmes
be x.
By using all the information in the table, the number of viewers in each region of the Venn
diagram can be filled as follows.
News
19 Total number viewers = 120
21 + 16 + x + 27 – x + 19 + 13 + 41 – x = 120
27 – x 13 137 – x = 120
x x = 17
21 41 – x
Sports 16 Entertainment
Hence, the number of viewers who watched sports or news and also entertainment programmes
= 16 + x + 13
= 16 + 17 + 13
= 46
Answer: C
Pointers
• The region which represents the viewers who watched
News
19
sports or news and also entertainment programmes is 13
the shaded region shown in the Venn diagram.
Sports 16 Entertainment
• By finding the number of viewers represented by the intersection of the three sets, the
problem can be solved.
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4. NEXUS SPM MATHEMATICS FORM 4 & 5
Page 41 (More Cloned SPM Questions) Supplementary Materials for
Cloned SPM Question (2006, Paper 2)
The Venn diagram below shows the sets P, Q and R such that the universal set ξ = P ∪ Q ∪ R.
P R Q
On separate diagrams, shade
(a) the set P' ∩ Q,
(b) the set (P' ∪ Q') ∩ R.
Solution
(a) P' ∩ Q
P R Q
(b) (P' ∪ Q') ∩ R
P R Q
Pointers
• P' ∩ Q means the complement of P intersects with Q, that is the region outside P and
overlapping with Q.
• (P' ∪ Q') ∩ R can be considered as the region of (P' ∪ Q') intersecting or overlapping
with R.
• The region of (P' ∪ Q') can be represented by the shaded region as shown in the Venn
diagram below.
P Q
As a result, (P' ∪ Q') ∩ R is R.
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