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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The Statement/PropositionGame
3 + 4 = 8
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? falsefalse
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The Statement/PropositionGame
y > 5
Is thisastatement?Is thisastatement? yesyes
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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The Statement/PropositionGame
x + y > 10
Is thisastatement?Is thisastatement? yesyes
Is thisa proposition?Is thisa proposition? nono
What is the truth valueWhat is the truth valueof the proposition?of the proposition? No truth valueNo truth value
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The Statement/PropositionGame
Isit raining?
Is thisastatement?Is thisastatement? nono
Is thisa proposition?Is thisa proposition? nono
** It hasno truth value.** It hasno truth value.
(Itsaquestion.)(Itsaquestion.)
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The Statement/PropositionGame
If the worldisround, then Columbus wasright.
Is thisastatement?Is thisastatement? yesyes
Is thisa proposition?Is thisa proposition? yesyes
What is the truth valueWhat is the truth value
of the proposition?of the proposition? probably trueprobably true
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The Statement/PropositionGame
x < y ifand only if y > x.
Is thisastatement?Is thisastatement? yesyes
Is thisa proposition?Is thisa proposition? yesyes
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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THEEND