MATHEMATICAL REASONING

STATEMENT

A SENTENCE EITHER TRUE OR FALSE BUT NOT BOTH

STATEMENT

TEN IS LESS THAN ELEVEN STATEMENT ( TRUE )

TEN IS LESS THAN ONE STATEMENT ( FALSE)

PLEASE KEEP QUIET IN THE LIBRARY NOT A STATEMENT

nono SentenceSentence statemestatementnt

NotNot

statemenstatementt

reasonreason

11 123 is 123 is divisible by divisible by 33

22

33 X-2 X-2 ≥ 9≥ 944 Is 1 a prime Is 1 a prime

number?number?

55 All octagons All octagons have eight sideshave eight sides

543 22

true

false

Neither true or false

A question

true

QUANTIFIERS

USED TO INDICATE THE QUANTITYALL – TO SHOW THAT EVERY OBJECT

SATISFIES CERTAIN CONDITIONS

SOME – TO SHOW THAT ONE OR MORE OBJECTS SATISFY CERTAIN CONDITIONS

QUANTIFIERS

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

OPERATIONS ON SETS

NEGATION The truth value of a statement can be changed

by adding the word “not” into a statement.

TRUE FALSE

NEGATION

EXAMPLE

P : 2 IS AN EVEN NUMBER ( TRUE )

P (NOT P ) : 2 IS NOT AN EVEN NUMBER

(FALSE )

COMPOUND STATEMENT

COMPOUND STATEMENT

A compound statement is formed when two statements are combined by using

“Or” “and”

COMPOUND STATEMENT

PP QQ P AND QP AND Q

TRUETRUE TRUETRUE TRUETRUE

TRUETRUE FALSEFALSE FALSEFALSE

FALSEFALSE TRUETRUE FALSEFALSE

FALSEFALSE FALSEFALSE FALSEFALSE

COMPOUND STATEMENT

PP QQ P OR Q P OR Q

TRUETRUE TRUETRUE TRUETRUE

TRUETRUE FALSEFALSE TRUETRUE

FALSEFALSE TRUETRUE TRUETRUE

FALSEFALSE FALSEFALSE FALSEFALSE

COMPOUND STATEMENT

EXAMPLE :

P : All even numbers can be divided by 2 ( TRUE )Q : -6 > -1 ( FALSE )

P and Q :

FALSE

COMPOUND STATEMENT

P : All even numbers can be divided by 2 ( TRUE )Q : -6 > -1 ( FALSE )

P OR Q :

TRUE

IMPLICATIONS

SENTENCES IN THE FORM

‘ If p then q ’ , where

p and q are statements

And p is the antecedent

q is the consequent

IMPLICATIONS

Example :

If x3 = 64 , then x = 4 Antecedent : x3 = 64 Consequent : x = 4

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

“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

“p if and only if q”

“p if and only if q”

If p , then q If p , then q If q , then p

Homework !!!!

Pg: 96 No 1 and 2

Pg: 98 No 1, 2 ( b, c ) 4 ( a, b, c, d)

IMPLICATIONS

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

IMPLICATIONS

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

ARGUMENTS

Mathematical reasoning

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

ARGUMENTS

Example : All quadrilaterals have four sides. A rhombus

is a quadrilateral. Therefore, a rhombus has four sides.

ARGUMENTS

There are three forms of arguments :

Argument Form I ( Syllogism )Premise 1 : All A is B

Premise 2 : C is A

Conclusion : C is B

ARGUMENTSArgument 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

ARGUMENTS

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

ARGUMENTS

Example

Premise 1 : If x = 6 , then x + 4 = 10Premise 2 : x = 6Conclusion : x + 4 = 10

ARGUMENTS

Argument Form III (Modus Tollens )Premise 1 : If p , then qPremise 2 : Not q is trueConclusion : Not p is true

ARGUMENTS

Example : Premise 1 : If ABCD is a square, then ABCD

has four sidesPremise 2 : ABCD does not have four sides.Conclusion : ABCD is not a square

ARGUMENTS

Completing the arguments

recognise the argument form

Complete the argument according to its form

ARGUMENTS

Example Premise 1 : All triangles have a sum of

interior angles of 180Premise 2 : ___________________________Conclusion : PQR has a sum of interior

angles of 180

PQR is a triangle

Argument Form I

ARGUMENTS

Premise 1 : If x - 6 = 10 , then x = 16

Premise 2 :__________________________

Conclusion : x = 16

Argument Form II

x – 6 = 10

ARGUMENTS

Premise 1 : __________________________

Premise 2 : x is not an even number

Conclusion : x is not divisible by 2

Argument Form III

If x divisible by 2 , then x is an even number

ARGUMENTS

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

DEDUCTION AND

INDUCTION

MATHEMATICAL REASONING

REASONING

There are two ways of making conclusions through reasoning by

a) Deduction b) Induction

DEDUCTION

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

DEDUCTION

Example :

All students in Form 4X are present today.David is a student in Form 4X.Conclusion : David is present today

general

Specific

INDUCTION

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

INDUCTIONINDUCTION

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 .

REASONING

Deduction

Induction

GENERAL SPECIFIC