chapter 4 - mathematical reasoning form 4

7/29/2019 Chapter 4 - Mathematical Reasoning Form 4

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MATHEMATICAL

REASONING


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STATEMENT

A SENTENCE EITHERTRUE ORFALSE BUT NOT BOTH


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STATEMENT

TEN IS LESS THAN ELEVEN

STATEMENT ( TRUE )

TEN IS LESS THAN ONE STATEMENT ( FALSE)

PLEASE KEEP QUIET IN THE LIBRARY

NOT A STATEMENT


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STATEMENT

no Sentence statement Not

Statementreason

1 123 is

divisible by 3

2 3 + 2 = 4 3 X-2 9 4 Is 1 a prime

number?

5 All octagons haveeight sides

true

false

Neither true or false

A question

true


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QUANTIFIERS

USED TO INDICATE THE QUANTITY

ALL TO SHOW THAT EVERY OBJECTSATISFIES CERTAIN CONDITIONS

SOME TO SHOW THAT ONE OR MOREOBJECTS SATISFY CERTAIN CONDITIONS


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QUANTIFIERS

EXAMPLE :

- All cats have four legs

- Some even numbers are divisible by 4- All perfect squares are more than 0


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OPERATIONS ON SETS

NEGATION

The truth value of a statement can be changed by

adding the word not into a statement.

TRUE FALSE


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NEGATION

EXAMPLE

P : 2 IS AN EVEN NUMBER ( TRUE )

P (NOT P ) : 2 IS NOT AN EVEN

NUMBER (FALSE )


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COMPOUND STATEMENT


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COMPOUND STATEMENT

A compound statement is formed when twostatements are combined by using

Or and


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COMPOUND STATEMENT

P Q P AND Q

TRUE TRUE TRUE

TRUE FALSE FALSE

FALSE TRUE FALSE

FALSE FALSE FALSE


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COMPOUND STATEMENT

P Q P OR Q

TRUE TRUE TRUETRUE FALSE TRUE

FALSE TRUE TRUE

FALSE FALSE FALSE


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COMPOUND STATEMENT

EXAMPLE :

P : All even numbers can be divided by 2

( TRUE )Q : -6 > -1

( FALSE )

P and Q :FALSE


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COMPOUND STATEMENT

P : All even numbers can be divided by 2

( TRUE )

Q : -6 > -1

( FALSE )

P OR Q :

TRUE


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IMPLICATIONS

SENTENCES IN THE FORM

If p then q ,where

p and q are statements

And p is the antecedent

q is the consequent


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IMPLICATIONS

Example :

Ifx3 = 64 , then x = 4

Antecedent : x3 = 64Consequent : x = 4


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IMPLICATIONS

Example :

Identify the antecedent and consequent for the implication

below.

If the weather is fine this evening, then I will playfootball

Answer :

Antecedent : the weather is fine this evening

Consequent : I will play football


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p if and only ifq

The sentence in the form p if and only ifq , is acompound statement containing two implications:

a) Ifp , then q

b) Ifq, thenp


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p if and only ifq

p if and only ifq

If p , then q If q , then p


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IMPLICATIONS

The converse of

If p ,then qis

if q , then p.


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IMPLICATIONS

Example :

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


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ARGUMENTS

Mathematical reasoning


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ARGUMENTS

What is argument ?

- A process of making conclusion based on a set ofrelevant information.

- Simple arguments are made up of two premises anda conclusion


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ARGUMENTS

Example :

All quadrilaterals have four sides. A rhombus is a

quadrilateral. Therefore, a rhombus has four sides.


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ARGUMENTS

There are three forms of arguments

:


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ARGUMENTS

Argument Form I ( Syllogism )

Premise 1 : All A is B

Premise 2 : C is A

Conclusion : C is B


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ARGUMENTS

Argument Form 1( Syllogism )

Make a conclusion based on the premises givenbelow:

Premise 1 : All even numbers can be dividedby 2

Premise 2 : 78 is an even number

Conclusion : 78 can be divided by 2


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ARGUMENTS

Argument Form II ( Modus Ponens ):

Premise 1 : If p , then q

Premise 2 : p is true

Conclusion : q is true


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ARGUMENTS

Example

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

Premise 2 : x = 6Conclusion : x + 4 = 10


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ARGUMENTS

Argument Form III (Modus Tollens )

Premise 1 : If p , then q

Premise 2 : Not q is true

Conclusion : Not p is true


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ARGUMENTS

Example :

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

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


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ARGUMENTS

Completing the arguments

recognise the argument form

Complete the argument according to its form


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ARGUMENTS

Example

Premise 1 : All triangles have a sum of interiorangles of 180

Premise 2 :___________________________

Conclusion : PQR has a sum of interiorangles of 180

PQR is a triangle

Argument Form I


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ARGUMENTS

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

Premise 2:__________________________

Conclusion : x = 16

Argument Form II

x 6 = 10


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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 evennumber


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D E D U C T I O N

A N DI N D U C T I O N

MATHEMATICAL

REASONING


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REASONING

There are two ways of making conclusions throughreasoning by

a) Deductionb) Induction


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DEDUCTION

IS A PROCESS OF MAKING A

SPECIFIC CONCLUSION BASED ON AGIVEN GENERAL STATEMENT


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DEDUCTION

Example :

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

Conclusion : David is present today

general

Specific


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INDUCTION

A PROCESS OF MAKING A GENERAL

CONCLUSION BASED ON SPECIFIC CASES.


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INDUCTION


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INDUCTION

Amy is a student in Form 4X. Amy likesPhysics

Carol is a student in Form 4X. Carol likesPhysics

Elize is a student in Form 4X. Elize likesPhysics

..

Conclusion : All students in Form 4X likePhysics .


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REASONING

Deduction

Induction

GENERAL SPECIFIC

chapter 4 - mathematical reasoning form 4

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