1. MATHEMATICAL
REASONING

2. STATEMENT
A SENTENCE EITHER TRUE OR
FALSE BUT NOT BOTH

3. STATEMENT
TEN IS LESS THAN ELEVEN
STATEMENT ( TRUE )
TEN IS LESS THAN ONE
STATEMENT ( FALSE)
PLEASE KEEP QUIET IN THE LIBRARY
NOT A STATEMENT

4. no Sentence statement Not reason
statement
1 123 is true
divisible by 3
2
3 +4 =5
2 2 false
3 X-2 ≥ 9 Neither true or false
4 Is 1 a prime
A question
number?
5 All octagons have
true
eight sides

5. QUANTIFIERS
USED TO INDICATE THE QUANTITY
ALL – TO SHOW THAT EVERY OBJECT
SATISFIES CERTAIN CONDITIONS
SOME – TO SHOW THAT ONE OR MORE
OBJECTS SATISFY CERTAIN CONDITIONS

6. QUANTIFIERS
EXAMPLE :
- All cats have four legs
- Some even numbers are divisible by 4
- All perfect squares are more than 0

7. OPERATIONS ON SETS
NEGATION
The truth value of a statement can be changed by
adding the word “not” into a statement.
TRUE FALSE

8. NEGATION
EXAMPLE
P : 2 IS AN EVEN NUMBER ( TRUE )
∼P (NOT P ) : 2 IS NOT AN EVEN
NUMBER (FALSE )

9. COMPOUND STATEMENT

10. COMPOUND STATEMENT
A compound statement is formed when two
statements are combined by using
 “Or”
 “and”

11. COMPOUND STATEMENT
P Q P AND Q
TRUE TRUE TRUE
TRUE FALSE FALSE
FALSE TRUE FALSE
FALSE FALSE FALSE

12. COMPOUND STATEMENT
P Q P OR Q
TRUE TRUE TRUE
TRUE FALSE TRUE
FALSE TRUE TRUE
FALSE FALSE FALSE

13. COMPOUND STATEMENT
EXAMPLE :
P : All even numbers can be divided by 2
( TRUE )
Q : -6 > -1
( FALSE )
P and Q :
FALSE

14. COMPOUND STATEMENT
P : All even numbers can be divided by 2
( TRUE )
Q : -6 > -1
( FALSE )
P OR Q :
TRUE

15. IMPLICATIONS
 SENTENCES IN THE FORM
‘ If p then q ’ ,
where
p and q are statements
And p is the antecedent
q is the consequent

16. IMPLICATIONS
Example :
If x3 = 64 , then x = 4
Antecedent : x3 = 64
Consequent : x = 4

17. IMPLICATIONS
Example :
Identify the antecedent and consequent for the implication
below.
“ If the weather is fine this evening, then I will play
football”
Answer :
Antecedent : the weather is fine this evening
Consequent : I will play football

18. “p if and only if q”
The sentence in the form “p if and only if q” , is a
compound statement containing two implications:
a) If p , then q
b) If q , then p

19. “p if and only if q”
“p if and only if q”
If p , then q If q , then p

20. Homework !!!!
Pg: 96 No 1 and 2
Pg: 98 No 1, 2 ( b, c )
4 ( a, b, c, d)

21. IMPLICATIONS
The converse of
“If p ,then q”
is
“if q , then p”.

22. IMPLICATIONS
Example :
If x = -5 , then 2x – 7 = -17

23. Mathematical reasoning
ARGUMENTS

26. ARGUMENTS
What is argument ?
- A process of making conclusion based on a set of
relevant information.
- Simple arguments are made up of two premises and
a conclusion

27. ARGUMENTS
Example :
All quadrilaterals have four sides. A rhombus is a
quadrilateral. Therefore, a rhombus has four sides.

28. ARGUMENTS
There are three forms of
arguments :

29. Argument Form I ( Syllogism )
Premise 1 : All A is B
Premise 2 : C is A
Conclusion : C is B

30. ARGUMENTS
Argument Form 1( Syllogism )
Make a conclusion based on the premises given
below:
Premise 1 : All even numbers can be divided
by 2
Premise 2 : 78 is an even number
Conclusion : 78 can be divided by 2

31. ARGUMENTS
Argument Form II ( Modus Ponens ):
Premise 1 : If p , then q
Premise 2 : p is true
Conclusion : q is true

32. ARGUMENTS
Example
Premise 1 : If x = 6 , then x + 4 = 10
Premise 2 : x = 6
Conclusion : x + 4 = 10

33. ARGUMENTS
Argument Form III (Modus Tollens )
Premise 1 : If p , then q
Premise 2 : Not q is true
Conclusion : Not p is true

34. ARGUMENTS
Example :
Premise 1 : If ABCD is a square, then ABCD
has four sides
Premise 2 : ABCD does not have four sides.
Conclusion : ABCD is not a square

35. ARGUMENTS
Completing the arguments
 recognise the argument form
Complete the argument according to its form

36. ARGUMENTS
Example
Premise 1 : All triangles have a sum of interior
angles of 180°
Premise 2 : PQR is a triangle
___________________________
Conclusion : PQR has a sum of interior
angles of 180°
Argument Form I

37. ARGUMENTS
Premise 1 : If x - 6 = 10 , then x = 16
Premise 2 x – 6 = 10
:__________________________
Conclusion : x = 16
Argument Form II

38. ARGUMENTS
If x divisible by 2 , then x is an even
Premise 1 : __________________________
number
Premise 2 : x is not an even number
Conclusion : x is not divisible by 2
Argument Form III

39. ARGUMENTS
Homework :
Pg : 103 Ex 4.5 No 2,3,4,5

40. MATHEMATICAL
REASONING
DEDUCTION
AND
INDUCTION

43. REASONING
There are two ways of making conclusions through
reasoning by
a) Deduction
b) Induction

44. DEDUCTION
IS A PROCESS OF MAKING A
SPECIFIC CONCLUSION BASED ON A
GIVEN GENERAL STATEMENT

45. DEDUCTION
Example :
general
All students in Form 4X are present today.
David is a student in Form 4X.
Conclusion : David is present today
Specific

46. INDUCTION
A PROCESS OF MAKING A GENERAL
CONCLUSION BASED ON SPECIFIC CASES.

47. INDUCTION

48. INDUCTION
Amy is a student in Form 4X. Amy likes
Physics
Carol is a student in Form 4X. Carol likes
Physics
Elize is a student in Form 4X. Elize likes
Physics
……………………………………………………..
Conclusion : All students in Form 4X like
Physics .

49. REASONING
Deduction
GENERAL SPECIFIC
Induction