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SPM Maths

3.1 Set

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3.1 Set
1. A set is a collection of objects according to certain characteristics
2. The objects in a set are known as elements.
3. Sets are usually denoted by capital letters and notation used for sets is braces, {   }.

Example:
A = {1, 3, 5, 7, 9}
 
4. In set notation, the symbol ∈ means ‘is an element of’ or ‘belongs to’ and ∉ means ‘is not an element of’ or ‘does not belong to’.
 
Example 1:
Given that P= {factors of 15} and Q = {positive perfect squares less than 28}. By using the symbol ∈ or ∉ , complete each of the following:
(a) 5 ___  P   (b) 20 ___ P   (c) 25 ___ Q   (d) 8  ___ Q
 
Solution:
P = {1, 3, 5, 15}, Q = {1, 4, 9, 16, 25}

(a) 5 ∈ P ← 5 is an element of set P (b) 20 ∉ P ← 20 is not an element of set P (c) 2 5 ∈ Q ← 2 5 is an element of set Q (d) 8 ∉ Q ← 8 is not an element of set Q



(A) Represent sets by using Venn diagram
5. A set can be represented by a Venn diagram using closed geometry shapes such as circles, rectangles, triangles and etc.

6. A dot to the left of an object in a Venn diagram indicates that the object is an element of the set.

7. When a Venn diagram represents the number of elements in a set, no dot is placed to the left of the number.
 
Example 2:
(a) Draw a Venn diagram to represent each of the following sets.
(b) State the number of elements for each of the set.
A= {2, 3, 5, 7}
B= {k, m, r, t, y}
 
Solution:
(a)
(b)
n(A) = 4
n(B) = 5



(B) Determine whether a set is an empty set
8. A set with no elements is called an empty set or null set. The symbol φ or empty braces, {  }, denotes empty set.
For example, if set A is an empty set, then A = {  } or A = φ and
n (A) = 0.
 
9. If B = {0} or {φ} does not denote that B is an empty set. B = {0} means that there is an element ‘0’ in set B.
B = {φ} means that there is an element ‘φ’ in set B.

3.4 SPM Practice (Short Questions)

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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 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:

  1. X’ ∩ Y means the intersection of the region outside X with the region Y.
    • Find the region of (X ∪ Y’) first.
    • (X ∪ 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 ∪ Y’) ∩ Z.

Question 2:

The Venn diagrams in the answer space shows sets P, Q and R such that the universal set,
On the diagrams in the answer space, shade

  1. Q ∩ R,
  2. (P’ ∩ R) ∪ Q.

Solution:

  1. Q ∩ R means the intersection of the region Q and the region R.
    • 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) ∪ Q.

 

 

 

3.3 Operations on Sets

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3.3b Union of Sets (Part 1)
 
1. The union of set A and set B, denoted by A υ B is the set consisting of all elements in set A or set B or both the sets.
The Venn diagram of A υ B is illustrated as below:

 

2. The union of set A, set B and set C, denoted by A υ B υ C is the set consisting of all elements in set A, set B or set C or all the three sets.
The Venn diagram of A υ B υ C is illustrated as below:


 

3.2 Subset, Universal Set and the Complement of a Set

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3.2c Complement of a Set
1. The complement of set B is the set of all elements in the universal set, ξ, which are not elements of set B, and is denoted by B’.

Example 1:
If ξ = {17, 18, 19, 20, 21, 22, 23} and
B = {17, 20, 21} then
B’ = {18, 19, 22, 23}

2. The Venn diagram below shows the relationship between B, B’ and the universal set, ξ.









The complement of set B is represented by the green colour shaded region inside the universal set, ξ, but outside set B.

3.3 Operations on Sets (Part 1)

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3.3a Intersection of Sets
 
1. The intersection of set P and set Q, denoted by P ∩ Q  is the set consisting of all elements common to set P and set Q.

2. The intersection of set P, set Q and set R, denoted by P ∩ Q ∩ R  is the set consisting of all elements common to set P, set Q and set R.

3. Represent the intersection of sets using Venn diagrams.

 (a) P ∩ Q





(b) Q ⊂ P , then P ∩ Q = Q

(c) P ∩ Q = ∅ , There is no intersection between set P and set Q .

(d) P ∩ Q ∩ R




3.2 Subset, Universal Set and the Complement of a Set

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3.2a Subsets
1. If every element of a set A is also an element of a set B, then set A is called subset of set B.

2. The symbol ⊂ is used to denote ‘is a subset of’.
Therefore, set A is a subset of set B. In set notation, it is written as A ⊂ B.

Example:
A = {11, 12, 13} and B = {10, 11, 12, 13, 14}
Every element of set A is an element of set B.
Therefore  A ⊂ B .

3.   A ⊂ B .  can be illustrated using Venn diagram as below:


4. The symbol   ⊄ is used to denote ‘is not a subset of’.

5. An empty set is a subset of any set.
For example, ∅ ⊂ A

6. A set is a subset of itself.
For example,   B ⊂ B  

7. The number of subsets for a set with n elements is 2n.
For example, if A = {3, 7}
So n = 2, then number of subsets of set A = 22 = 4
All the subsets of set A are { }, {3}, {7} and {3, 7}.

3.2 Subset, Universal Set and the Complement of a Set

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3.2b Universal Set
1. Universal set is a set that contains all the elements under consideration.
2. In set notation, the symbol ξ denotes a universal set.

Example:
Given that the universal set, ξ = {whole numbers less than 9}, A = {prime number} and B = {multiple of 4}.
(a) List all the elements of set A and set B.
(b) Illustrate the relationships between the following sets using Venn diagrams.
(i) ξ and A
(ii) ξ, A and B

Solution:
(a) ξ = {0, 1, 2, 3, 4, 5, 6, 7, 8}
 A = {2, 3, 5, 7}
 B = {4, 8}
 
(b)(i)








(b)(ii)

3.4 SPM Practice (Short Questions)

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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 ∪ 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 ∪ 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∪C)' = {5, 6, 7, 8, 10, 14}
Therefore  (A∪C)’ ∩ B = {5, 7}

 

 

 

2.3 Quadratic Equations

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2.3 Quadratic Equations
1. Quadratic equations are equations which fulfil the following characteristics:
(a) Have an equal ‘=’ sign
(b) Contain only one unknown
(c) Highest power of the unknown is 2.
 
For example,

 
2. The general form of a quadratic equation is written as:
(a) 
ax2 + bx + c = 0,
where a ≠ 0, b ≠ 0 and c ≠ 0,
example: 4x2+ 13x – 12 = 0

(b) 
ax2 + bx = 0,
where a ≠ 0, b ≠ 0 but c = 0,
example: 7x2+ 9x = 0

(c) 
ax2 + c = 0,
where a ≠ 0, c ≠ 0 but b = 0,
example: 9x2– 3 = 0



Example 1:
Write each quadratic equation in the general form.
(a) x2 – 5x = 12
(b) -2 + 5x2– 6x = 0
(c) 7p2 – 3p = 4p2+ 4p – 3
(d) (x – 2)(x + 6) = 0
(e) 3 – 13x = 4 (x2 + 2)
(f)   2 − y = 1 − 3 y y
(g)  p 4 = 2 p 2 − 3 10
(h) y 2 + 5 4 = y − 1 2
(i) 4 p 7 = p ( 7 p − 6 )


Solution:
A quadratic equation in the general form is written as ax2 + bx + c = 0
(a) x2 – 5x = 12
x2 – 5x -12 = 0

(b) –2 + 5x2– 6x = 0
5x2 – 6x –2 = 0


(c) 7p2 – 3p = 4p2+ 4p – 3
7p2 – 3p – 4p2– 4p + 3 = 0
3p2 – 7p  + 3 = 0

(d) (x – 2)(x + 6) = 0
x2 + 6x – 2x– 12 = 0
x2 + 4x – 12 = 0

(e) 3 – 13x = 4 (x2 + 2)
3 – 13x = 4x2 + 8
–4x2 – 8 + 3 – 13x = 0
–4x2 – 13x – 5 = 0
4x2 + 13x + 5 = 0

(f) 2 − y = 1 − 3 y y
2y – y2 = 1 – 3y
2y – y2 – 1 + 3y = 0
 – y2 + 5y – 1 = 0
y2 – 5y + 1 = 0


(g)  p 4 = 2 p 2 − 3 10
10p = 8p2 – 12
–8p2 + 10p +12 = 0
8p2 – 10p – 12 = 0

(h) y 2 + 5 4 = y − 1 2
2y2 + 10 = 4y – 4
2y2 – 4y + 10 + 4 = 0
2y2 – 4y + 14 = 0

(i)  4 p 7 = p ( 7 p − 6 )
4p = 7p (7p– 6)
4p = 49p2 – 42p
– 49p2 + 42p + 4p  = 0
49p2 – 46p = 0