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4.2 Quantifiers ‘All’ and ‘Some’ (Sample Questions)

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Example 1:
Complete each of the following statements using the quantifiers ‘all’ or ‘some’ to make the statement true.
(a) _____ rectangles are squares.
(b) _____ prime numbers are odd numbers.
(c) _____ triangles have equal sides.
(d) _____ even numbers are divisible by 2.

Answer:
(a) Some rectangles are squares.
(b) Some prime numbers are odd numbers.
(c) Some triangles have equal sides.
(d) All even numbers are divisible by 2.



Example 2:
Construct a true statement using the quantifier ‘all’ or ‘some’ for the given object and property.
(a) Object: multiples of 4
 Property: can be divided exactly by 5
(b) Object: regular hexagon
 Property: 6 equal sides
(c) Object: acute angles
 Property: less than 90o

Answer:
(a) Some multiples of 4 can be divided exactly by 5.
(b) All regular hexagons have 6 equal sides.
(c) All acute angles are less than 90o.

4.5 Arguments (Short Notes)

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4.5 Arguments

(A) Premises and Conclusions
1. An argument is a process of making conclusion based on several given statements.
2. The statements given are known as premises.
3. An argument consists of premises and a conclusion.

Example 1:
Identify the premises and conclusion of the following argument.
(a) A pentagon has 5 sides. ABCDE is a pentagon. Therefore, ABCDE has 5 sides.

Solution:
Premise 1: A pentagon has 5 sides.
Premise 2: ABCDE is a pentagon.
Conclusion: ABCDE has 5 sides.


(B) Forms of Arguments
1. Based on two given premises, a conclusion can be made for three different forms of arguments.

Argument Form I
Premise 1: All A are B.
Premise 2: C is A.
Conclusion: C is B.
Example 2:
Make a conclusion based on the two premises given below.
Premise 1: All multiples of 5 are divisible by 5.
Premise 2: 45 is a multiple of 5.
Conclusion: ______________

Solution:
Conclusion: 45 is divisible by 5.

Argument Form II
Premise 1: If p, then q.
Premise 2: p is true.
Conclusion: q is true.
Example 3:
Make a conclusion based on the two premises given below.
Premise 1: If a number is a factor of 18, then the number is a factor of 54.
Premise 2: 3 is a factor of 18.
Conclusion: ______________

Solution:
Conclusion: 3 is a factor of 54.

Argument Form III
Premise 1: If p, then q.
Premise 2: Not q is true.
Conclusion: Not p is true.

Example 4:
Make a conclusion based on the two premises given below.
Premise 1: If P is a subset of Q, then P  ∩ Q = P.
Premise 2: P  ∩ Q  ≠ P
Conclusion: ______________

Solution:
Conclusion: P is not the subset of Q.

4.1 Statements

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(A) Determine Whether a Given Sentence is a Statement
1. A statement is a sentence which is either true or false but not both.

2. Sentences which are questions, instructions and exclamations are not statements.

Example 1:
Determine whether the following sentences are statements or not. Give a reason for your answer.
(a) 3 + 3 = 8
(b) A pentagon has 5 sides.
(c) Is 40 divisible by 3?
(d) Find the perimeter of a square with each side of 4 cm.
(e) Help!

Solution:
(a) Statement; it is a false statement.
(b) Statement; it is a true statement.
(c) Not a statement; it is a question.
(d) Not a statement; it is an instruction.
(e) Not a statement; it is an exclamation.



(B) Determine Whether a Statement is True or False.

Example 2:
Determine whether each of the following statements is true or false.
(a) 7 is a prime number
(b) -10 > -7
(c) 3 is a factor of 8.

Solution:
(a) True
(b) False
(c) False



(C) Constructing Statements Using Numbers and Symbols
 
1. True and false statements can be constructed with numbers and mathematical symbols.
 
Example 3:
Construct (i) a true statement, (ii) a false statement,
using the following numbers and mathematical symbols.
(a) 2, 4, 8, ×, =
(b) {a, b, c}, {d} , U =

Solution:
(a)(i) A true statement: 2 × 4 = 8
(a)(ii) A false statement: 2 × 8 = 4
(b)(i) A true statement: {d} U {a, b, c} = {a, b, c, d}
(b)(ii) A false statement: {d} U {a, b, c} = {d}



4.2 Quantifiers ‘All’ and ‘Some’

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4.2 Quantifiers ‘All’ and ‘Some’
 
Statement using ‘All’ and ‘Some’
1. Quantifiers are words that denote the number of objects or cases referred to in a given statement.
2. Quantifier ‘all’, ‘any’ and ‘every’ describe each and every object or case.
3. Quantifier ‘some’, ‘several’ and ‘part of’ describe one or more objects or cases.
 
Example:
Complete each of the following statements using the quantifiers ‘all’ or ‘some’ to make the statement true.
(a) _______  polygons have the same number of vertices and sides.
(b) _______  multiples of 9 are even numbers.
(c) _______  of the whole numbers are divisible by 7.
(d) _______  factors of 4 are factors of 20.
 
Solution:
(a) All polygons have the same number of vertices and sides.
(b) Some multiples of 9 are even numbers.
(c) Some of the whole numbers are divisible by 7.
(d) All factors of 4 are factors of 20.

4.3 Operations on Statements

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4.3 Operations on Statements (Part 1)
 
(A) Nagating a Statement using ‘No’ or ‘Not’
 
1. Negation of a statement refers to changing the truth value of the statement, that is, changing a true statement to a false statement and vice versa, using the word ‘not’ or ‘no’.
 
Example 1:
Change the true value of the following statements by using ‘no’ or ‘not’.
(a) 17 is a prime number.
(b) 39 is a multiple of 9.
 
Solution:
(a) 17 is not a prime number. (True to false)
(b) 39 not is a multiple of 9. (False to true)


2. A compound statement can be formed by combining two given statements using the word ‘and’.
 
Example 2:
Identify two statements from each of the following compound statements.
(a) All pentagons have 5 sides and 5 vertices.
(b) 33 = 27 and 43 = 64
 
Solution:
(a) All pentagons have 5 sides.
 All pentagons have 5 vertices.
(b) 33 = 27
  43 = 64



Example 3:
Form a compound statement from each of the following pairs of statements using the word ‘and’.
(a) 19 is a prime number.
  19 is an odd number.
(b) 15 – 5 = 10
  15 × 5 = 75

Solution:
(a) 19 is a prime number and an odd number. ← (Repeated words can be eliminated when combining two statements using ‘and’.)

(b) 15 – 5 = 10 and15 × 5 = 75.


3. A compound statement can also be formed by combining two given statements using the word ‘or’.
 
Example 4:
Form a compound statement from each of the following pairs of statements using the word ‘or’.
(a) 11 is an odd number.
  11 is a prime number.
(b) − 3 = − 27 3 − 3 = − 4 + 1

Solution:
(a) 11 is an odd number or a prime number.
( b) − 3 = − 27 3 or − 3 = − 4 + 1

3.4 SPM Practice (Long Questions)

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Question 3:
(a) The Venn diagrams in the answer space shows sets P and Q such that the universal set, ξ = P υ Q.
  Shade the set P ∩ Q.
(b) The Venn diagrams in the answer space shows sets X and Y and Z, such that the universal set, ξ = X υ Y υ Z.
  Shade the set (X υ Z) ∩ Y.

Solution:
(a)

• P ∩ Q means the intersection of the region P and the region Q.

(b)


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





Question 4:
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) P ∩ R’,
(b) P’υ (Q ∩ R).

Solution:
(a)
P ∩ R’
(b)
P’υ (Q ∩ R)


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.

3.2 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}.