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4.3 Operations on Statements (Sample Questions 2)

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Example 3:
Determine whether each of the following statements is true or false.
(a) 3 × (-4) = -12 and 13 + 6 = 19
(b) 100 × 0.7 = 70 and 12 + (-30) = 18

Solution:
When two statements are combined using ‘and’, a true compound statement is obtained only if both statements are true.
If one or both statements are false, then the compound statement is false.

(a) Both the statements ‘3 × (-4) = -12’ and ‘13 + 6 = 19’ are true. Therefore, the statement ‘3 × (-4) = -12 and 13 + 6 = 19’ is true.

(b) The statement ‘12 + (-30) = 18’ is false. Therefore, the statement ‘100 × 0.7 = 70 and 12 + (-30) = 18’ is false.



Example 4:
Determine whether each of the following statements is true or false.
(a) m + m = m2 or p × p × p = p-3
(b)  64 3 =−4 or  −27 3 =−3

Solution:
When two statements are combined using ‘or’, a false compound statement is obtained only if both statements are false.
If one or both statements are true, then the compound statement is true.

(a) Both the statements ‘m+ m = m2’ and ‘p × p × p= p-3’ are false. Therefore the statement m + m = m2 or p × p × p = p-3 is false.

(b) The statement  ' − 27 3 = − 3 '  is true. Therefore, the statement  64 3 =−4 or  −27 3 =−3  is true.

4.3 Operations on Statements

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4.3 Operations on Statements (Part 3)

(C) Truth Values of Compound Statements using ‘Or’
 
1. When two statements are combined using ‘or’, a false compound statement is obtained only if both statements are false.
 
2. f one or both statements are true, then the compound statement is true.
 
The truth table:
Let p = statement 1 and q= statement 2.
The truth values for ‘p’ or ‘q’ are as follows:


 
Example 6:
Determine the truth value of the following statements.
(a) 60 is divisible by 4 or 9.
(b) 53 = 25 or 43 = 64.
(c) 5 + 7 > 14 or √9 = 2.
 
Solution:
(a)
60 is divisible by 4  ← (p is true)
60 is divisible by 9  ← (q is false)
Therefore, 60 is divisible by 4 or 9 is a true statement. (‘p or q’ is true)
 
(b)
53 = 25  ← (p is false)
43 = 64  ← (q is true)
Therefore, 53 = 25 or 43 = 64 is a true statement. (‘p or q’ is true)
 
(c)
5 + 7 > 14  ← (p is false)
√9 = 2  ← (q is false)
Therefore, 5 + 7 > 14 or √9 = 2 is a false statement. (‘p or q’ is false)

4.3 Operations on Statements

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4.3 Operations on Statements (Part 2)
(B) Truth Values of Compound Statements using ‘And’
 
4. When two statements are combined using ‘and’, a true compound statement is obtained only if both statements are true.
 
5. If one or both statements are false, then the compound statement is false.
 
The truth table:
Let p = statement 1 and q = statement 2.
The truth values for ‘p’ and ‘q’ are as follows:

 
Example 5:
Determine the truth value of the following statements.
(a) 12 × (–3) = –36 and 15 – 7 = 8.
(b) 5 > 3 and –4 < –5.
(c) Hexagons have 5 sides and each of the interior angles is 90o.
 
Solution:
(a)
12 × (–3) = –36 ← (p is true)
15 – 7 = 8 ← (q is true)
Therefore 12 × (–3) = –36 and 15 – 7 = 8 is a true statement. (‘p and q’ is true)
 
(b)
5 > 3 ← (p is true)
–4 < –5 ← (q is false)
Therefore 5 > 3 and –4 < –5 is a false statement. (‘p and q’ is false)

(c)
Hexagons have 5 sides. ← (p is false)
Each of the interior angles of Hexagon is 90o. ← (q is false)
Therefore Hexagons have 5 sides and each of the interior angles is 90o is a false statement. (‘p and q’ is false)

3.4 SPM Practice (Long Questions)

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Question 7:
The Venn diagrams 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 set
(a) P ∩ Q,
(b) P ' ∩Q∪R


Solution:
(a) 
P ∩ Q


(b) 
P ' ∩Q∪R



Question 8:
The Venn diagrams 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 set
(a) Q ' ∩R
(b) ( Q∪R )∩ P '



Solution:
(a) 
Q ' ∩R


(b) 
( Q∪R )∩ P '



3.4 SPM Practice (Long Questions)

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Question 5:
The Venn diagrams 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) P ∩ Q,
(b) Q∩( P ' ∪R).

Solution:
(a) 
P ∩ Q

(b)
Q∩( P ' ∪R).




Question 6:
The Venn diagrams 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) P ∩ Q,
(b) P∩ (Q∪R) ' .

Solution:
(a)
P ∩ Q


(b)
P∩ (Q∪R) ' .



3.4 SPM Practice (Long Questions)

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Question 1:
The Venn diagrams in the answer space shows sets X, Y and Z such that the universal set, ξ = X ∪ Y ∪ Z
On the diagrams in the answer space, shade
( a ) X' ∩ Y , ( b ) ( X ∪ Y' ) ∩ Z


Solution:
(a) 
• X’ ∩ Y means the intersection of the region outside X with the region Y.

 
 
(b)
• Find the region of (X υ Y’) first.
• (X υ Y’) means the union of the region X and the region outside Y.
• The region then intersects with region Z to give the result of (X υ Y’) ∩ Z.





Question 2:
The Venn diagrams in the answer space shows sets P, Q and R such that the universal set, ξ = P υ Q υ R
On the diagrams in the answer space, shade
(a) Q ∩ R,
(b) (P’ ∩ R) υ Q.

Solution:
(a) 
• Q ∩ R means the intersection of the region Q and the region R.



(b)
• Find the region of (P’ ∩ R) first.
• (P’ ∩ R) means the region that is outside P and is inside R.
• The union of this region with region Q give the result of (P’ ∩ R) υ Q.



3.4.2 Sets, SPM Paper 1 (Short Questions 5 – 7)

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3.4.2 Sets, SPM Practice Paper 1 (Short Questions)

Question 5:
Diagram below shows a Venn diagram with the number of elements of set P, set Q and set R.

It is given that the universal set, ξ=P∪Q∪R and n( Q' )=n( Q∩R ).
Find the value of x.

Solution:
n(Q)’ = n(Q ∩ R)
3 + 8 + 5 = x– 3 + 9
16 = x + 6
x = 10


Question 6:
Diagram below is a Venn diagram showing the number of quiz participants in set P, set Q and set R.
It is given that the universal set, ξ = P ∪ Q ∪ R , set P = {Science quiz participants}, set Q = {Mathematics quiz participants} and set R = {History quiz participants}.

If the number of participants who participate in only one quiz is 76, find the total number of the participants.

Solution:
Number of participants who participate in only one quiz = 76
(5x – 2) + (x + 6) + (2x + 8) = 76
8x + 12 = 76
8x = 64
x = 8
Total number of the participants
= 76 + 7 + 4 + 5 + 3(8)
= 116


Question 7:
Diagram below is a Venn diagram showing the number of students in set K, set L and set M.
It is given that the universal set, ξ = K ∪ L ∪ M , set K = {Karate Club}, set L = {Life Guards Club} and set M = {Martial Arts Club}.

If the number of students who join both the Life Guards Club and the Martial Arts Club is 8, find the number of students who join only two clubs.

Solution:
Number of students who join both the Life Guards Club and the Martial Arts Club = n(L ∩ M) = 2 + 2x
2 + 2x = 8
2x = 6
x = 3
Number of students who join only two clubs
= x + 4 + 2x
= 3 + 4 + 2(3)
= 13

3.4.1 Sets, SPM Paper 1 (Short Questions 1 – 4)

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Question 1:
List all the subsets of set P = {r, s}.
 
Solution:
There are 2 elements, so the number of subsets of set P is 2n = 22= 4.
Set P = {r, s}
Therefore subsets of set P = {r}, {s}, {r, s}, { }.


Question 2:
 
Diagram above shows a Venn diagram with the universal set, ξ = Q υ P.
List all the subset of set P.
 
Solution:
Set P has 3 elements, so the number of subsets of set P is 2n = 23 = 8.
Set P = {2, 3, 5}
Therefore subsets of set P = { }, {2}, {3}, {5}, {2, 3}, {2, 5}, {3, 5}, {2, 3, 5}.


Question 3:
It is given that the universal set, ξ = {x: 30 ≤ x < 42, x is an integer} and set P = {x: x is a number such that the sum of it its two digits is an even number}.
Find set P’.
 
Solution:
ξ = {30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41}
P = {31, 33, 35, 37, 39, 40}
Therefore P’ = {30, 32, 34, 36, 38, 41}.


Question 4:
Given that universal set ξ = {x : 3 < x ≤ 16, x  is an integer},
Set A = {4, 11, 13, 16},
Set B = {x : x is an odd number} and
Set C = {x : x is a multiple of 3}.

The elements of the set (A υ C)’ ∩ B are

Solution:
ξ = {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
A = {4, 11, 13, 16}
B = {5, 7, 9, 11, 13, 15}
C = {6, 9, 12, 15}
(A υ C)’ = {5, 7, 8, 10, 14}
Therefore (A υ C)’ ∩ B = {5, 7}.

3.3 Operations on Sets

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3.3b Union of Sets (Part 2)
Example 1:
The Venn diagram below shows the number of elements in the universal set, ξ, set P, set Q and R.


Given n(Q) = n(P υ R)’, find n(ξ).
 
Solution:
n(Q) = n (P υ R)’
2x + 6 + 1 + 5 = 2x + 2x
2x + 12 = 4x
2x = 12
x = 6
 
n(ξ) = 2x + 2x + x + 7 + 6 + 1 + 5
= 5x+ 19
= 5(6) + 19
= 30 + 19
= 49
 


Example 2:
Diagram below is a Venn diagram showing the universal set, ξ = {Form 3 students}, set A = {Students who play piano} and set B = {Students who play violin}.
 

 
Given n(ξ) = 60, n(A) = 25, n(B) = 12 and n(A ∩ B) = 8, find the number of students who do not play the two instruments.

Solution:
The students who do not play the two instruments are represented by the shaded region, (A υ B)’.

 
Number of students who do not play the two instruments
= n (A υ B)’
= 60 – 17 – 8 – 4
= 31
 

3.3a Intersection of Sets (Part 2)

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3.3a Intersection of Sets
Example 1:
Given that A= {3, 4, 5, 6, 7}, B = {4, 5, 7, 8, 9, 12} and C = {3, 5, 7, 8, 9, 10}.
(a) Find A∩B∩C.
(b) Draw a Venn diagram to represent A∩B∩C.
 
Solution:
(a)  A∩B∩C= {5, 7}

(b)


 

4. The complement of the intersection of two sets, P and Q, represented by (P ∩ Q)’, is a set that consists of all the elements of the universal set, ξ, but not the elements of P ∩ Q.

5. The complement of set (P ∩ Q)’ is represented by the shaded region as shown in the Venn diagram.