MATHEMATICAL MATHEMATICAL REASONINGREASONING

STATEMENTSTATEMENT

A SENTENCE EITHER A SENTENCE EITHER TRUETRUE OR OR FALSE FALSE BUT NOT BOTH BUT NOT BOTH

STATEMENTSTATEMENT

TEN IS LESS THAN ELEVEN TEN IS LESS THAN ELEVEN STATEMENTSTATEMENT ( TRUE ) ( TRUE )

TEN IS LESS THAN ONE TEN IS LESS THAN ONE STATEMENTSTATEMENT ( FALSE) ( FALSE)

PLEASE KEEP QUIET IN THE LIBRARYPLEASE KEEP QUIET IN THE LIBRARY NOT A STATEMENTNOT 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

QUANTIFIERSQUANTIFIERS

USED TO INDICATE THE QUANTITYUSED TO INDICATE THE QUANTITY ALLALL – TO SHOW THAT – TO SHOW THAT EVERY OBJECTEVERY OBJECT

SATISFIES CERTAIN CONDITIONSSATISFIES CERTAIN CONDITIONS

SOMESOME – TO SHOW THAT – TO SHOW THAT ONE OR ONE OR MOREMORE OBJECTS SATISFY CERTAIN OBJECTS SATISFY CERTAIN CONDITIONSCONDITIONS

QUANTIFIERSQUANTIFIERS

EXAMPLE :EXAMPLE :

- All cats have four legsAll cats have four legs- Some even numbers are divisible by Some even numbers are divisible by

44- All perfect squares are more than 0All perfect squares are more than 0

OPERATIONS ON SETSOPERATIONS ON SETS

NEGATION NEGATION

The truth value of a statement can be The truth value of a statement can be

changed by adding the word “changed by adding the word “notnot” ” into a statement.into a statement.

TRUE FALSE TRUE FALSE

NEGATIONNEGATION

EXAMPLEEXAMPLE

P : 2 IS AN EVEN NUMBER ( TRUE )P : 2 IS AN EVEN NUMBER ( TRUE )

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

))

COMPOUND COMPOUND STATEMENTSTATEMENT

COMPOUND STATEMENTCOMPOUND STATEMENT

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

“ “Or”Or” “ “and”and”

COMPOUND STATEMENTCOMPOUND STATEMENT

PP QQ P AND QP AND Q

TRUETRUE TRUETRUE TRUETRUE

TRUETRUE FALSEFALSE FALSEFALSE

FALSEFALSE TRUETRUE FALSEFALSE

FALSEFALSE FALSEFALSE FALSEFALSE

COMPOUND STATEMENTCOMPOUND STATEMENT

PP QQ P OR Q P OR Q

TRUETRUE TRUETRUE TRUETRUE

TRUETRUE FALSEFALSE TRUETRUE

FALSEFALSE TRUETRUE TRUETRUE

FALSEFALSE FALSEFALSE FALSEFALSE

COMPOUND STATEMENTCOMPOUND STATEMENT

EXAMPLE :EXAMPLE :

P : All even numbers can be divided by P : All even numbers can be divided by 2 2

( TRUE )( TRUE )Q : -6 > -1Q : -6 > -1 ( FALSE )( FALSE )

P P andand Q : Q :

FALSEFALSE

COMPOUND STATEMENTCOMPOUND STATEMENT

P : All even numbers can be divided P : All even numbers can be divided by 2 by 2

( TRUE )( TRUE )

Q : -6 > -1Q : -6 > -1

( FALSE )( FALSE )

P P OROR Q : Q :

TRUETRUE

IMPLICATIONSIMPLICATIONS SENTENCES IN THE FORMSENTENCES IN THE FORM

‘ ‘ IfIf pp thenthen q q ’ ,’ , wherewhere

pp and and qq are statements are statements

And And p is the antecedentp is the antecedent

q is the consequentq is the consequent

IMPLICATIONSIMPLICATIONS

Example :Example :

If If xx33 = 64 = 64 , then , then x = 4x = 4

Antecedent : Antecedent : xx33 = 64 = 64

Consequent : Consequent : x = 4x = 4

IMPLICATIONSIMPLICATIONS

Example :Example : Identify the antecedent and consequent for the Identify the antecedent and consequent for the

implication below.implication below.

“ “ If the whether is fine this evening, If the whether is fine this evening, then I will play football”then I will play football”

Answer :Answer : Antecedent : the whether is fine this eveningAntecedent : the whether is fine this evening

Consequent : I will play footballConsequent : I will play football

““pp if and only if if and only if qq””

The sentence in the form “The sentence in the form “pp if and if and only ifonly if qq” , is a compound statement ” , is a compound statement containing containing twotwo implications: implications:

a) If a) If pp , , then then qq

b) If b) If qq , then , then pp

““pp if and only if if and only if qq””

“ “pp if and only if if and only if qq””

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

Homework !!!!Homework !!!!

Pg: 96 No 1 and 2Pg: 96 No 1 and 2

Pg: 98 No 1, 2 ( b, c )Pg: 98 No 1, 2 ( b, c )

4 ( a, b, c, d)4 ( a, b, c, d)

IMPLICATIONSIMPLICATIONS

The converse ofThe converse of

“ “If p ,then q” If p ,then q”

is is

“ “if q , then p”.if q , then p”.

IMPLICATIONSIMPLICATIONS

Example :Example :

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

Mathematical reasoningMathematical reasoning

ArgumentsArguments

ARGUMENTS ARGUMENTS

What is argument ?What is argument ?- A process of making conclusion A process of making conclusion

based on a set of relevant based on a set of relevant information.information.

- Simple arguments are made up of Simple arguments are made up of two premises and a conclusiontwo premises and a conclusion

ARGUMENTSARGUMENTS

Example :Example :

All quadrilaterals have four sides. A All quadrilaterals have four sides. A rhombus is a quadrilateral. rhombus is a quadrilateral. Therefore, a rhombus has four sides.Therefore, a rhombus has four sides.

ARGUMENTSARGUMENTS

There are There are threethree forms of forms of arguments :arguments :

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

Premise 2 : C is APremise 2 : C is A

Conclusion : C is BConclusion : C is B

ARGUMENTSARGUMENTSArgument Form 1( SyllogismArgument Form 1( Syllogism ) ) Make a conclusion based on the premises given Make a conclusion based on the premises given

below:below: Premise 1 : All even numbers can be divided Premise 1 : All even numbers can be divided

by 2 by 2 Premise 2 : 78 is an even numberPremise 2 : 78 is an even number

Conclusion : 78 can be divided by 2Conclusion : 78 can be divided by 2

ARGUMENTSARGUMENTS

Argument Form II ( Modus Argument Form II ( Modus Ponens ):Ponens ):

Premise 1Premise 1 : If : If pp , then , then qq

Premise 2Premise 2 : p is true : p is true

ConclusionConclusion : q is true : q is true

ARGUMENTSARGUMENTS

Example Example

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

Premise 2Premise 2 : x = 6 : x = 6

ConclusionConclusion : x + 4 = 10 : x + 4 = 10

ARGUMENTSARGUMENTS

Argument Form III (Modus Argument Form III (Modus Tollens )Tollens )

Premise 1Premise 1 : If p , then q : If p , then q

Premise 2Premise 2 : Not q is true : Not q is true

ConclusionConclusion : Not p is true : Not p is true

ARGUMENTSARGUMENTS

Example : Example :

Premise 1Premise 1 : If ABCD is a square, then : If ABCD is a square, then ABCD ABCD has four sides has four sides

Premise 2Premise 2 : ABCD does not have four : ABCD does not have four sides.sides.

ConclusionConclusion : ABCD is not a square : ABCD is not a square

ARGUMENTSARGUMENTS

Completing the argumentsCompleting the arguments

recognise the argument formrecognise the argument form

Complete the argument according to Complete the argument according to its formits form

ARGUMENTSARGUMENTS

Example Example

Premise 1 : All triangles have a sum of Premise 1 : All triangles have a sum of interior interior angles of 180 angles of 180

Premise 2 : ___________________________Premise 2 : ___________________________

Conclusion : PQR has a sum of interior Conclusion : PQR has a sum of interior angles of 180 angles of 180

PQR is a triangle

Argument Form I

ARGUMENTSARGUMENTS

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

Premise 2 :__________________________Premise 2 :__________________________

Conclusion : x = 16 Conclusion : x = 16

Argument Form II

x – 6 = 10

ARGUMENTSARGUMENTS

Premise 1 : __________________________Premise 1 : __________________________

Premise 2 : x is not an even numberPremise 2 : x is not an even number

Conclusion : x is not divisible by 2Conclusion : x is not divisible by 2

Argument Form III

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

ARGUMENTSARGUMENTS

Homework :Homework :

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

MATHEMATICAL MATHEMATICAL REASONINGREASONING

DEDUCTION DEDUCTION

AND AND

INDUCTIONINDUCTION

REASONINGREASONING

There are two ways of making There are two ways of making conclusions through reasoning by conclusions through reasoning by

a) Deductiona) Deduction

b) Inductionb) Induction

DEDUCTIONDEDUCTION

IS A PROCESS OF MAKING A IS A PROCESS OF MAKING A

SPECIFIC CONCLUSION BASED ON A SPECIFIC CONCLUSION BASED ON A

GIVEN GENERAL STATEMENT GIVEN GENERAL STATEMENT

DEDUCTIONDEDUCTION

Example : Example :

All students in Form 4X are present All students in Form 4X are present today.today.

David is a student in Form 4X.David is a student in Form 4X.

Conclusion : David is present todayConclusion : David is present today

general

Specific

INDUCTIONINDUCTION

A PROCESS OF MAKING A GENERAL A PROCESS OF MAKING A GENERAL

CONCLUSION BASED ON SPECIFIC CONCLUSION BASED ON SPECIFIC CASES.CASES.

INDUCTIONINDUCTION

INDUCTIONINDUCTION

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 .

REASONINGREASONING

DeductionDeduction

InductionInduction

GENERAL SPECIFIC