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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.

4.4 Implication

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(A) Antecedent and Consequent of an Implication

  1. For two statements p and q, the sentence ‘if p, then q’ is called an implication.
  2. p is called the antecedent.
    q is called the consequent.


Example:

Identify the antecedent and consequent of the following implications.

(a) If m = 2, then 2m2 + m = 10

(b) I f P ∪ Q = P , t h e n Q ⊂ P

Solution:

  1. Antecedent: m = 2
    Consequent: 2m2 + m = 10
  2. Antecedent : P ∪ Q = P Consequent : Q ⊂ P

(B) Implications of the Form ‘p if and only if q’

  1. Two implications ‘if p, then q’ and ‘if q, then p’ can be written as ‘p if and only if q’.
  2. Likewise, two statements can be written from a statement in the form ‘p if and only if q’ as follows:
    Implication 1: If p, then q.
    Implication 2: If q, then p.


Example 1:

Given that p: x + 1 = 8
q: x = 7
Construct a mathematical statement in the form of implication

  1. If p, then q.
  2. p if and only if q.

Solution:

  1. If x + 1 = 8, then x = 7.
  2. x + 1 = 8 if and only if x = 7.

Example 2:

Write down two implications based on the following sentence:
x3 = 64 if and only if x = 4.

Solution:

If x3 = 64, then x = 4.
If x = 4, then x3 = 64.

(C) Converse of an Implication

  1. The converse of an implication ‘if p, then q’ is ‘if q, then p’.

Example:

State the converse of each of the following implications.

  1. If x2 + x – 2 = 0, then (x – 1)(x + 2) = 0.
  2. If x = 7, then x + 2 = 9.

Solution:

  1. If (x – 1)(x + 2) = 0, then x2 + x – 2.
  2. If x + 2 = 9, then x = 7.

 

 

4.6 SPM Practice (Long Questions)

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Question 7:
(a)(i) State whether the following compound statement is true or false.

3 + 3 = 9 or   3 × 3 = 9
(a)(ii) Determine whether the following converse is true or false.

If x > 3, then x > 7
(b) Write down Premise 2 to complete the following argument:
Premise 1: If y = mx + 5 is a linear equation, then m is a gradient of the straight line.
Premise 2: _____________________
Conclusion: 2 is the gradient of the straight line.

(c) Angle subtended at the centre of a regular polygon with n sides is 360 o n .
Make one conclusion by deduction for the angle subtended at the centre of a regular polygon with 5 sides.

Solution:
(a)(i) True

(a)(ii) The converse is true

(b) Premise 2: y = 2x + 5 is a linear equation

(c)
Angle subtended at the centre of a regular pentagon = 360 o n = 360 o 5 = 72 o


Question 8:
(a) State whether the following sentence is a statement or not a statement.

x + 7 = 9
(b) Complete the following compound statement by writing the word ‘or’ or ‘and’ to form a true statement.
23 = 6 …… 5 × 0 = 0

(c) Write down Premise 2 to complete the following argument:
Premise 1: All isosceles triangles have two equal sides.
Premise 2: _____________________
Conclusion: ABC has two equal sides.

(d) Make a general conclusion by induction for the sequence of numbers 1, 7, 17, 31, … which follows the following pattern.
1 = (2 × 1) – 1
7 = (2 × 4) – 1
17 = (2 × 9) – 1
31 = (2 × 16) – 1

Solution:
(a) Not a statement

(b) 23 = 6 …or… 5 × 0 = 0

(c) ABC is an isosceles triangle.

(d)
1 = (2 × 1) – 1 =  (2 × 12) – 1
7 = (2 × 4) – 1 =  (2 × 22) – 1
17 = (2 × 9) – 1 =  (2 × 32) – 1
31 = (2 × 16) – 1 =  (2 × 42) – 1
  =  2n2 – 1, n = 1, 2, 3, …

4.5 Arguments

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(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.

  1. 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.

 

 

4.6 SPM Practice (Long Questions)

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Question 5:
(a) State if each of the following statements is true or false.
 (i)    23= 8 or ⅓ = 1.33.
 (ii)  – 6 > – 8 and 6 > 8.
 
(b) Write down two implications based on the following statement:

x a + y b =1 if and only if bx+ay=ab.

(c) It is given that the interior angle of a regular polygon of n sides 
  is   ( 1 − 2 n ) × 180 ∘ .
  Make one conclusion by deduction on the size of the interior angle of a regular hexagon.

Solution:
(a)(i) True

(a)(ii) False

(b) 
Implication 1:   If   x a + y b =1, then bx+ay=ab. _ Implication 2:   If  bx+ay=ab, then  x a + y b =1. _

(c)
Size of an interior angle of a regular hexagon = ( 1 − 2 6 ) × 180 ∘ = 2 3 × 180 ∘ = 120 ∘


Question 6:
(a) Complete the following mathematical sentence by writing the symbol > or <.
  (i) 53____ 20 is a false statement.
  (ii) – 3 ____ – 10 is a true statement.

(b) Complete the conclusion in the following argument:
 Premise 1 : If  n 1 2 = n , then  4 1 2 = 4 =2.   Premise 2 :  n 1 2 = n  Conclusion : _____________________
(c) Make a general conclusion by induction for the sequence of numbers 10, 35, 70, … which follows the following pattern.

10 = 5 (2)2 – 10 
35 = 5 (3)2 – 10
70 = 5 (4)2 – 10
…. = ………..

Solution:
(a)(i) 53   <    20 is a false statement.

(a)(ii)   – 3    >    – 10 is a true statement.

(b)
Conclusion : 4 1 2 = 4 = 2

(c) 5 (n + 1)2 – 10, where n = 1, 2, 3, …


4.4 Implications Short Notes

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4.4 Implications

(A) Antecedent and Consequent of an Implication
1. For two statements p and q, the sentence ‘if p, then q’ is called an implication.

2. p is called the antecedent.
   q is called the consequent.

Example:
Identify the antecedent and consequent of the following implications.
(a) If m = 2, then 2m2 + m = 10
(b) If P∪Q=P, then Q⊂P

Solution:
(a) Antecedent: m = 2
Consequent:: 2m2 + m = 10

(b) Antecedent:P∪Q=P       Consequent:Q⊂P


(B) Implications of the Form ‘p if and only if q’
1. Two implications ‘if p, then q’ and ‘if q, then p’ can be written as ‘p if and only if q’.
2. Likewise, two statements can be written from a statement in the form ‘p if and only if q’ as follows:
Implication 1: If p, then q.
Implication 2: If q, then p.

Example 1:
Given that p: x + 1 = 8
 q: x = 7
Construct a mathematical statement in the form of implication
(a) If p, then q.
(b) p if and only if q.

Solution:
(a) If x + 1 = 8, then x = 7.
(b) x + 1 = 8 if and only if x = 7. 

Example 2:
Write down two implications based on the following sentence:
x3 = 64 if and only if x = 4.

Solution:
If x3 = 64, then x = 4.
If x = 4, then x3 = 64.



(C) Converse of an Implication
1. The converse of an implication ‘if p, then q’ is ‘if q, then p’.
 
Example:
State the converse of each of the following implications.
(a) If x2 + x – 2 = 0, then (x - 1)(x + 2) = 0.
(b) If x = 7, then x + 2 = 9.

Solution:
(a) If (x - 1)(x + 2) = 0, then x2 + x – 2.
(b) If x + 2 = 9, then x = 7.

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)