Ma Thematic A Reasoning

8/6/2019 Ma Thematic A Reasoning

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Mathematical

REASONING


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Statements

The Basic Unit Involved In

Mathematical Reasoning Is A

Mathematical Statement.

For Example,

A Crow Is a bird.

Sun Is a Planet .


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We Can Understand That The First

Sentence Is True And The SecondOne Is Not True, As These Are

Known To Everyone, And Are

Sometimes Referred To As FACTS.

A SENTENCE IS CALLED A

MATHEMATICALLY

ACCEPTABLE STATEMENT IF IT

IS EITH

ERT

RUE OR FALSE. .


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Truth Value of a Statement:

A statement is said to have truth valueTorF ,according as the statement

considered is true or false. Forexample, the statement 2 plus 2 is fourhas truth value T, whereas, thestatement 2 plus 2 is five has truthvalue F. The knowledge of truth valueof statements enables to replace onestatement by some other (equivalent)

statement(s).


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ConsiderThe Following

Statements:

Mathematics Is Fun.

How Far Is Chennai From Here?

These Sentences Are Not Always

True. It Varies With The SituationAnd May Vary For DifferentPeople.


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Production of new

statements

Negation of a statement.

Compound statement.

Implication.

Conjunctions. Disjunctions.


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Negation of a statement

Ifp is a statement then its negation p

is statement not p, p has truth value

ForTaccording as the truth value

of p is TorF.

For example,

Plants Have Life.

The Negation OfThe Statement is ,

Plants Doesnt Have life.


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Examples for Negation of a Statement:Examples for Negation of a Statement:

Example 1: Every square has four sides.

Solution:

Let P Every square has four sides.

Here the statement P is the true statement. Therefore, the negationof the statement P is false

P Every square has four sides. ~P Every Square does not have four sides.

Example 2:

The number 6 is odd

Solution: Let Q The number 6 is odd

Here the statement Q is false statement. Therefore, the negation ofthe statement Q is true.

Q The number 6 is odd

~Q The number 6 is not odd


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Compound

Statements A Compound Statement Is A

Statement Which Is Made Up Of TwoOr More Statements. In This Case,

Each Statement Is Called A

Component Statement.


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Examples and Symbolic Representation of

Compound Statements of Maths

Examples of Compound statements

1. X10

2. P>50 and p6

0then send him for the testThe connectives used for AND, OR, IF THEN and IF AND ONLY

IF are ^, V, ----> and

Symbolic representation of compound statements

Let x: read the booksY: clears the exam

X ^ Y: x reads the books and y clears the exam

X v Y: x reads the books or y clears the exam

X--->Y: if x reads the books then y clears the exam

XY: X reads the books if and only if Y clears the exam


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Types of Compound Statements of Mathwith theirTruth Tables

1. Conjunction: ---if x and y are two statements then the compoundstatement x and y is called a conjunction denoted by x ^ y which istrue when both x and y are true

Truth table for conjunction

x y x^y T F F

T T T

2. Disjunction: ----if x and y are two statements then the compoundstatement x or y is called a disjunction denoted by x V y which is false

only if both x and y are false

Truth table for Disjunction

X Y xVy

T F T

T T T


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Examples OfCompoundStatements

THESKY IS BLUEAND THEGRASS IS GREEN.

TheComponentstatementsare:

P:The Sky Is Blue.

Q:The grass Is Green.

IT IS RAINING AND IT ISCOLD.

TheComponentStatementsare:

P:It Is Raining.

Q:It Is Cold.


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examples1. A cat is an

animal (true) and2 + 2 = 4 (true).

(T + T = T)

Since both facts

are true, the

entire sentence is

true.

Remember: For a

compoundstatement using

"and" to be true,

BOTH facts must be

true


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The word or

Two lines in A Plane Are Either Intersecting Or

parallel.

There Are Two Types Of ORstatements:

EXCLUSIVE OR

INCLUSIVE OR


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Exclusive OR

Put differently, exclusive disjunction is a logicaloperation on two logical values, typically the values oftwo propositions, that produces a value oftrue only in

cases where the truth value of the operands differ. For Example,

A Stundent Who took Bio Or Chem Can Apply For M.Sc.

The logical operation exclusive disjunction, also calledexclusive or(symbolized XOR, EOR, EXOR, or


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Venn diagram of A Bbut not is

Graphical representation

Venn diagram of

A B C


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INCLUSIVE OR In logic and mathematics, or, also known as

logical disjunction orinclusive disjunction, isa logical operator that results in true wheneverone or more of its operands are true. E.g. in thiscontext, "A orB" is true ifA is true, or ifB is true,

or if bothA and B are true. In grammar, oris acoordinating conjunction. In ordinary language"or" sometimes has the meaning of exclusivedisjunction. For Example,

An Ice Cream Or Pepsi is Available With ALunch.

Oris usually expressed with an infix operator. Inmathematics and logic, it is usually V.


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Graphical representation

Venn diagram of A V B


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QUANTIFIERS These Are Phrases Like,There Exist And for

All.

For Example,

There Exist A Rectangle Whose

All Sides Are Equal. Another Word is For Every

For Example,

For Every Prime Number p, Root

p is Irrational.

These Two words AreCalledQuantifiers.


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Implications Here We Discuss The Implications Of If-Then,only If

and If And Only If.

For Example,

If You Are Born In Some Country, then You

Are A Citizen Of That Country.

p:You Are Born In Some Country.

q: You Are Citizen Of That Country.

Here not Happening Of p Has No Effect On NotHappening Of q.


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CONTRAPOSITIVE and

converse These Are Certain Statements Which Can Be FormedFrom A Given Statement With If-Then.

For Example,

If The Physical Environment Changes, Then theBiological Environment Changes.

ThenCONTRAPOSITIVEOf The Statement Is,

If The Physical Environment Does Not Change, Then

The Biological Environment Does not Change.


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OTHER EXAMPLES

If A Number IS Divisible By 9,Then It Is Divisible by 3.

If You Are Born In India, Then You Are A Citizen of India.

If A Triangle Is Equilateral, It Is Isosceles.

THE CONTRAPOSITIVESARE,

If A Number Is Not Divisible by 9,Then It Is Not Divisibleby 3.

If You Are Not Born In India, Then You Are Not A Citizen

Of India.

If A Triangle Is Not Equilateral, It IS Isosceles.


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Converse Converse Of given Statement Ifp,Then q Is If

q,Thenp. For Example,

p:If A Number n Is Even ,Then n^2 Is Even.

q:If A Number n^2

Is Even ,Then n Is Even.


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Validating statements

Rule 1: If p And q Are Mathematical Statements,Thenin Order to show that the statement p and q is True, theFollowing Steps Are Followed:

STEP1:Show That Statement p Is True.

STEP2:Show That Statement q Is True.


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Rule 3: Statements With If-then

In Order To Prove the Statement Ifp then

q We Need To Show That Any One OfThe Following Case Is true.

Case 1: ByAssuming That p Is True,

prove That Q Must be True.

Case 2: ByAssuming That q Is False,

Prove That p Must be False.


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Rule 4: Statements With if And Only If

In Order to Prove The Statement p If

And Only If q,We Need To Show,

1)If PIs True,Then Q is true and

2)If q IsTrue,t

hen PIs

True.


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By Contradiction

We Check Whether A Statementp is true, We

Assume That p iS not true ie. ~p is true. Then,We Arrive At Some Result Which Contradicts

Our assumption.Therefore,We Conclude Thatp

Is True.

ForExample,

Proving That Root(a) Is Irrational Using the

Method Of Contradiction.


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Counter example

This Method Involves Giving a

Counter Example,ie. A Situation

Where The Situation Is Not Valid.

However, generating examples InFavour Of A Statement Do Not

Provide Validity OfThe statement.


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Historical note

The First Treatise on LogicWas writtenBy Aristotle. It Was A Collection Of

Rules For Deductive ReasoningW

hichWould Serve As A Basis For Study OfEvery Branch Of Knowledge. Later In17th Century, Leibnitz Conceived TheIdea Of Using Symbols In Logic toMechanize The Process Of DeductiveReasoning. LaterHis Ideas Were Put ToUse By De Morgan And George Boole.


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Ma Thematic A Reasoning