lecture chapter 1

8/3/2019 Lecture Chapter 1

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CHAPTER 1

LOGIC


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Learning outcomes

At theend of this topic, studentsshould beable to:

identify a proposition.

tell thename of the logical connectives construct truth table.

combine the propositions by using the logicalconnectives.

to show that a propositionis tautology /contradiction / equivalent.

identify law of logic.


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History

Greek philosopher.

Knownas the fatherof logic.

his works contain theearliest knownformal study oflogic, which were

incorporatedin the latenineteenthcentury into modernformal logic.

Born on Nov. 2, 1815 inLincoln, England. Died onDec. 8, 1864 inBallintemple,

Irelandat 49 yrs old.

In 1854, heestablished therules of

symbolic logic in his book TheLaws of

Thought.George Boole

AristotleAristotle


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Logic & Computer science

Someapplications of logic:

- Data base theory formalize the

definitions ofqueries.- Softwareengineering design of

electronic computersincluding thedesign

ofnetworks orcircuits.

- Programming languages to provea

program to be correct can use logic-based

notionssuch as loop invariants.


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The Statement/PropositionGame

Earth is the only planet in the universe that

contains life.

Is thisastatement?Is thisastatement? yesyes

Is thisa proposition?Is thisa proposition? yesyes

What is the truth valueWhat is the truth valueof the proposition?of the proposition? truetrue


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3 + 4 = 8



What is the truth valueWhat is the truth valueof the proposition?of the proposition? falsefalse


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y > 5


Is thisa proposition?Is thisa proposition? nono

Its truth valuedepends on thevalue of y andIts truth valuedepends on thevalue of y and

x, but thisvalueisnot specified.x, but thisvalueisnot specified.


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x + y > 10



What is the truth valueWhat is the truth valueof the proposition?of the proposition? No truth valueNo truth value


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Isit raining?

Is thisastatement?Is thisastatement? nono


** It hasno truth value.** It hasno truth value.

(Itsaquestion.)(Itsaquestion.)


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If the worldisround, then Columbus wasright.



What is the truth valueWhat is the truth value

of the proposition?of the proposition? probably trueprobably true


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x < y ifand only if y > x.



What is the truth valueWhat is the truth value

of the proposition?of the proposition?truetrue

( becauseits truth value doesnot depend on( becauseits truth value doesnot depend on

specific values of x and y.)specific values of x and y.)


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Test yourunderstanding on

proposition. Which of the following sentences are

proposition ?

i. Thesunisshining.

ii. nisa primenumber.

iii. Takean umbrella with you.iv. Letsgo to therestaurant.

v. Come to class!!

vi. Thesum of two primenumbersareeven.


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1.2 Logical Operators (Connectives) We formalize this by denoting propositions with letters

such as p, q, r, s, andintroducingseveral logicaloperators.

We will examine the following logical operators:

i) Negation (NOT)

ii) Conjunction (AND)iii) Disjunction (OR)

iv) ExclusiveOr(XOR)

v) Implication (if then)

vi) Biconditional (ifand only if)

Truth tables can be used to show how these operatorscan combine propositions to compound propositions.


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Negation (NOT)

Unary Operator, Symbol:

P P

true false

false true

e.g.

P = I am a Malaysian.

P = I am not a Malaysian


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Conjunction (AND)

Binary Operator, Symbol:

P Q P Q

true true true

true false false

false true false

false false false

e.g.P = I will have salad for lunch

Q = I will have fried mee for dinner .

P Q = I will have salad for lunch

and fried mee fordinner


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ExclusiveOr(XOR) Binary Operator, Symbol:

(p or q but not both)

P Q PQ

true true falsetrue false true

false true true

false false false

e.g.

P = I will order Fried rice fordinner

Q = I will order Chicken chop fordinner

P

Q = I will either order Friedrice or Chicken chop fordinner


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Implication (if - then)

Binary Operator, Symbol: p

P Q PpQ

true true true

true False false

false True true

false False true

e.g

P= you study hardQ= you will get good grade

P p Q= If you study hard thenyou will get good

grade


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Biconditional (ifand only if)

Binary Operator, Symbol: m

P Q PmQ

true true true

true false false

false true false

false false true

e.gP = You can take the flight

Q = You buy a ticket

P m Q = you can take the flight ifand only if you buy a ticket

P if Q and Q if PP if Q and Q if P


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Which is the correct truth table

forp q ?p q

T T T

T F T

F T T

F F F

p q

T T T

T F T

F T F

F F T

p q

T T T

T F F

F T T

F F T

p q

T T F

T F T

F T T

F F T

1. 2. 3. 4.

Hint: Lookat thevalues of p andq that make p q false


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1.3 Compound PropositionOne ormore propositions can be combined to form asinglecompound proposition.

Forexample:

Single propositions:

m : Fatimah isa pretty girl.

n : Fatimah isakindgirl.

Compound proposition:

m n : Fatimah isa pretty andkindgirl.


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Example

1. Show that thestatement forms (p q)and p qarenot logically equivalent.

2. Construct the truth table for thestatement

form (p q) (p q).

3. Write the followingsentencessymbolically,

letting h = It is hots = It issunny

a) It isnot hot andit issunny.

b) It iseitherhot orsunny.


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Example:4. If P= It is raining and

Q= Iam indoor

Use thesymbols of logical connectives torepresent the followingsentences:

i. It israiningand I am indoor.

ii. Ifit israining then I am indoor.

iii. It israiningif I am indoor.

iv. It israiningifand only if I am indoor.

v. It isnot raining.


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1.4 Tautologies and Contradictions

Tautology isastatement form which is true

forall values ofstatement variables.

Contradiction isastatement form which is false

forall values ofstatement variables.

Otherthan tautology & contradiction, the

propositionsare calledcontigency.


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1.5 TheLaws ofLogic1. Law ofDouble Negation2. De MorgansLaws

3. CommutativeLaws

4. AssociativeLaws

5. Idempotent Laws

6. Identity Laws

7. AbsorptionLaws

8. NegationLaws

9. DistributiveLaws

pp

qpqpqpqp

pqqppqqp

)()()()( rqprqprqprqp

pppppp

cqqtpp

ptppcp

pqpppqpp

)()()(

)()()(

rpqprqp

rpqprqp


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Simplify the followingstatement:

Solution

)()( qpqp

lawIdentityplawNegationcp

laweCommutativqqp

lawveDistributiqqp

lawnegativeDoubleqpqp

lawssMorganDeqpqpqpqp

)(

)(

)()(

')())(()()(


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Rules of Inference:

1. Modus ponens

p p q

If p, q.

e.g.

If password correct, then login permitted

__________________

Passwordis correct

Loginis permitted

2. Modus Tollen

p p q

If q, p.

e.g.

If password correct, then login permitted

______________

Loginnot permitted

Passwordincorrect.

3. EliminitionP q

p,

p, qe.g.

Either Amin is in the library or he isin the cafe .

_____________________Amin is not in the library.So he must be in the cafe.

4. Transitivitypp pp q

qq pp r pp pp r e.g.r e.g.If tomorrow is not a holiday then I have to goto work.If I have to go to work then I have to takethe train.

_____________________* If tomorrow is not a holiday then I have to

take the train.

If p is true thenqis true.

// p is truethereforeqis true

If p is true thenqis true.// qisnot truetherefore p is

not true

Eitherp orqis true.

// p isnot true, thereforeqis

true


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Example1. Recognizing modus Ponensand Modus Tollen

i) If Socratesisa man, then Socratesis mortal

Socratesisa man

.

ii) If thesum of thedigits of371,487isdivisible by 3, then371,487isdivisible by 3.

371,487isdivisible by 3.

iii) .

870,232 isnot divisible by 3.

870,232 isnot divisible by 6.


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2. Identify therules ofinference foreach of the following:

i)

If Anisahs fatheris Ahmad then Anisah could be my sister.**Anisah isnot my sister, therefore herfatherisnot Ahmad.

ii) If you get an A forthissubject then I will buy you ice cream.

** You scored A forthissubject. Therefore I will buy you icecream.

iii) x 3 = 0 or x + 2 = 0.

** x + 2 0

x 3 = 0

iv) If you invest in thestock market, then you will get rich.

v) If you get rich, then you will be happy.

** If you invest in thestock market, then you will be happy.


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Websites:

http://www.authorstream.com/Presentation/Silvia-46066-cs201-prop-logic-Data-Structures-Discrete-Mathematics-Propositional-logi-Education-ppt-powerpoint/ (dateaccessed: 3rd June 2009)

http://www.mathgoodies.com/lessons/toc_vol9.html (dateaccessed: 4thJune 2009)

http://www.mathwarehouse.com/math-statements/logic-and-truth-values.php (dateaccessed: 30th July 2009)

http://www.facstaff.bucknell.edu/mastascu/eLessonsHtml/Logic/Logic1.html

(dateaccessed: 5th Jan 2010)

http://www.jgsee.kmutt.ac.th/exell/Logic/Logic11.htm ( 12th Jan 2010)

(+ exercise recommended)


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lecture chapter 1

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