This document discusses various concepts in mathematical reasoning and logic: 1) It defines statements as sentences that are either true or false, and distinguishes them from questions, instructions, and exclamations. 2) It explains the quantifiers "all" and "some" and provides examples of their usage. 3) It demonstrates how to determine the truth value of compound statements using truth tables and discusses identifying implications, antecedents, and consequences in logical statements. 4) It describes the three main forms of arguments and provides examples of deduction and induction.

1. MATHEMATICAL REASONING

2.  A statement is a sentence which is either true or
false but not both.
 Sentences which are questions, instructions and
exclamations are not statements.
eg : 8 + 2 = 10 (statement)
4 × 5 = 9 (statement)
2p = 4 (not a statement)

3.  Write : i) a true statement and
ii) a false statement
involving the following numbers and
mathematical symbols.
eg : a) 1, 8 , 9, −, =
i) 9 − 8 = 1
ii) 1 − 9 = 8 or 8 − 9 = 1
or 1 − 8 = 9

4.  ‘All’ includes every object that fulfils a specific
requirement.
 ‘Some’ includes a certain number of objects that
fulfils a specific requirement under
consideration.
eg : a) Fill in the correct quantifier :
i) All pentagons have five sides.
ii) Some prime numbers can be
divided by 2.

5. eg :
b) Determine the truth of the following
statements.
i) All triangles have one of its internal angles
equal to 900
. [True statement]
ii) Some numbers are negative.
[True statement]
iii) Some quadrilaterals have four sides.
[False statement]

6.  Identifying two statements from compound
statement which contains the word ‘and’ or ‘or’
eg : a ) 2 and 7 are prime numbers.
Answer : 2 is prime number.
7 is prime number.
b) m + m + m = 3m or m × m = m
2
Answer : m + m + m = 3m
m × m = m
2

7.  Determining the truth value of a compound
statement containing the word ‘and’ or ‘or’.
Truth table :
eg : a) 10 - 3 = 7 and 10
2
= 20
(T) (F) = F
b) 10 - 3 = 7 or 10
2
= 20
(T) (F) = T
Statement 1 Statement 2 and or
T T T T
T F F T
F T F T
F F F F

8.  Constructing implications:
Eg: A polygon is a pentagon if and only if it has five sides.
Answer:
Implication 1: If a polygon has five sides, then it is a
pentagon.
Implication 2: If a polygon is a pentagon, then it has five
sides.
 Antecedent and consequence.
Eg: If 2x = 4, then x = 2.
Antecedent consequence.

9.  Three forms of arguments.
Form Argument Eg
1 Premise 1: All A are B.
Premise 2 : C is A.
Conclusion: C is B.
Premise 1: All isosceles triangles have two
equal sides.
Premise 2 : ABC is an isosceles triangle.
Conclusion: ABC has two equal sides.
2 Premise 1: If p, then q.
Premise 2: p is true.
Conclusion: q is true.
Premise 1: If m = 3, then m
3
= 27.
Premise 2 : m = 3.
Conclusion: m
3
= 27.
3 Premise 1: If p, then q.
Premise 2 : q is not true.
Conclusion: p is not true.
Premise 1: If m = 3, then m
3
= 27.
Premise 2 : m
3
≠ 27.
Conclusion: m ≠ 3.

10.  Deduction is the process of making specific
conclusion based on a general statement.
Eg :
Given Tn = ar
n − 1
, given a = 3, r = 2 and n = 4.
Answer :
T4 = (3)(2)
4 − 1
= 24

11.  Induction is the process of making a general
conclusion based on specific cases.
Eg : 2 = 4(1) − 2
6 = 4(2) − 2
10 = 4(3) − 2
…………..
Answer : 4n − 2, where n = 1, 2, 3, ………