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Mathematical Reasoning

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