Lecture Note Chap 01
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Transcript of Lecture Note Chap 01
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Dept. Mathematics, Kyung Hee Univ.
Discrete Mathematicswith Applications
LEE, SOOJOONDepartment of Mathematics
Kyung Hee University
Dept. Mathematics, Kyung Hee Univ.
Chapter 1 The Logic of Compound Statements
1.1 Logical Form & Logical Equivalence1.2 Conditional Statements
1.3 Valid and Invalid Arguments
Dept. Mathematics, Kyung Hee Univ.
1.1 Logical Form and Logical Equivalence
Dept. Mathematics, Kyung Hee Univ.
Statement (or Proposition)A sentence that is true or false but not both.
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Dept. Mathematics, Kyung Hee Univ.
Compound StatementsLogical operators
Negation:~ p (not p)Conjunction: p q (p and q) Disjunction: p q (p or q)
The order of logical operations~ p q = (~ p) q ~ (p q)p q r (p q) r, p (q r)
From English to symbolsIt is nothot butit is sunny ~ h sIt is neitherhot norsunny ~ h ~ s
Dept. Mathematics, Kyung Hee Univ.
Truth ValuesEither true (T) or false (F)Truth table
TFFT
~ pp
FFFFTFFFTTTT
p qqp
FFFTTFTFTTTT
p qqp
Dept. Mathematics, Kyung Hee Univ.
Statement (propositional) formstatement variables (such as p, q, r)+ logical connectives (such as ~, , )Logical operators
If and only if (iff)BiconditionIf thenCondition
Exclusive ORExclusive OR (XOR)OrDisjunctionAndConjunctionNot~Negation
MeaningSymbolName
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Truth Table for Exclusive ORp q = (p q ) ~ ( p q )
FFTFFTTT
(p q ) ~ ( p q )qp
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Dept. Mathematics, Kyung Hee Univ.
Truth Table for ( p q ) ~ r
FFFTFFFTFTTFFFTTFTFTTTTT
( p q ) ~ rrqp
Dept. Mathematics, Kyung Hee Univ.
Logical EquivalenceTwo statement forms are called logically equivalentif and only if they have identical truth table.Notation: PQExample: Double negation
~ (~ p) p
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Nonequivalence~ ( p q ) T ~ p ~ q~ ( p q ) T ~ p ~ q
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De Morgans Laws~ ( p q ) ~ p ~ q~ ( p q ) ~ p ~ q
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Dept. Mathematics, Kyung Hee Univ.
Tautologies & Contradictions Tautology
A statement form that is always true Contradiction
A statement form that is always false
Dept. Mathematics, Kyung Hee Univ.
p t p, p c ct : a tautology c : a contradiction
FFFFFT
p ccp
FTFTTT
p ttp
Dept. Mathematics, Kyung Hee Univ.
Logical Equivalences 1/3Commutative laws
p q q p, p q q pAssociative laws
(p q) rp (q r), (p q) rp (q r)Distributive laws
p (q r) (p q) (p r)p (q r) (p q) (p r)
Identity lawsp t p, p c p
Dept. Mathematics, Kyung Hee Univ.
Logical Equivalence 2/3Negation laws
p ~ pt, p ~ pcDouble negative law
~ (~ p) pIdempotent laws
p pp, p ppDe Morgans laws
~ (p q) ~ p ~ q~ (p q) ~ p ~ q
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Dept. Mathematics, Kyung Hee Univ.
Logical Equivalence 3/3Universal bound laws
p t t, p c cAbsorption laws
p (p q) p, p (p q) pNegation of t and c
~ t c, ~ c t
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ProblemsExercise 1.1 8, 9, 15, 28, 30, 39
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1.2 Conditional Statements
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p qIf p then q
p is called the hypothesisq is called the conclusionp is sufficient for qq is necessary for pp implies q
Truth TableFFTTTFTFF
TTTp qqp
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Dept. Mathematics, Kyung Hee Univ.
Truth table for p ~ q ~ p
T T TFFF T TTFT F FFTT F FTT
p ~ q ~ pqp
hypothesis conclusion
Dept. Mathematics, Kyung Hee Univ.
(p q) r(p r) (q r)
T T TF TFFFT T TF TTFFT F FT FFTFT T TT TTTFF F TT FFFTT T TT TTFTF F FT FFTTT T TT TTTT
(pr) (qr)p q rrqp
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p q~ p q
TTFFTTTFFFFTTTTT
~ p qp qqp
Dept. Mathematics, Kyung Hee Univ.
~ (p q) p ~ qThe negation of if p then q is logically equivalent to p and not q.~ (p q) ~ (~ p q) p ~ q
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Dept. Mathematics, Kyung Hee Univ.
ContrapositiveThe contrapositiveof p q is ~ q ~ pp q~ q ~ p
p q~ p q ~ (~ q) ~ p ~ q ~ p
TTFFTTTFFFFTTTTT
~ q ~ pp qqp
Dept. Mathematics, Kyung Hee Univ.
Converse & InverseThe converseof p q is q pThe inverseof p q is ~ p ~ q
TTTFFFFTTFTTFFTTTTTT
~ p ~ qq pp qqp
Dept. Mathematics, Kyung Hee Univ.
Only If & the biconditionalp only ifq means p q.The biconditionalof p and q is p if and only if q. (Notation: p q)p q( p q ) ( q p )
T T TT F FF F TT T T
( p q )( q p )
FFTFTFTFF
TTTp qqp
Dept. Mathematics, Kyung Hee Univ.
Necessary & Sufficient Conditionsp is a sufficient conditionfor q means p q.p is a necessary conditionfor q means ~ p ~ q.p is a necessary and sufficient conditionfor q means p q and ~ p ~ q.p q( p q ) ( q p )
( p q ) (~ p ~ q)
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Dept. Mathematics, Kyung Hee Univ.
ProblemsExercise 1.2 8, 11, 20, 23, 40, 41
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1.3 Valid and Invalid Arguments
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ArgumentA sequence of statementsPremises(or assumptionsor hypotheses) + ConclusionValid argument
If the resulting premises are all true, then the conclusion is also true.
Invalid argument (Fallacious argument)Not valid argument
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ExampleArgument
If Socrates is a man then Socrates is mortal.Socrates is a man. Socrates is mortal.
p: Socrates is a man, q: Socrates is mortalAbstract form
p qp q
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Dept. Mathematics, Kyung Hee Univ.
Invalid Argument Formp q ~ rq p r p r
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Valid Argument Formp (q r)~ r p q
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SyllogismSyllogism = Two premises + conclusion
1st premise (major premise)2nd premise (minor premise) Conclusion
Modus ponens (Method of affirming)p qp q
Modus tollens (Method of denying)p q~ q ~ p
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Modus Ponens & Modus Tollens
FFTTp
TTTFTFFTFTTTTFFTFFFFTFFTTTTT
~ p~ qp qqp qqp
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Dept. Mathematics, Kyung Hee Univ.
Additional Valid Argument Form : Rules of InferenceGeneralization
p p qq p q
Specializationp q pp q q
Eliminationp q, ~ q pp q, ~ p q
Transitivityp q, q r p r
Proof by Division into Casesp q, p r, q r r
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Complex DeductionPremises:
If my glasses are on the kitchen table, then I saw them at breakfast.I was reading the newspaper in the living room or I was reading the newspaper in the kitchen.If I was reading the newspaper in the living room, then my glasses are on the coffee table.I did not see my glasses at breakfast.If I was reading my book in bed, then my glasses are on the bed table.If I was reading the newspaper in the kitchen, then my glasses are on the kitchen table.
Where are the glasses?
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Symbolizing a situationPremises
(a) p q (b) r s (c) r t (d) ~ q (e) u v (f) s pPremises
(1) p q, ~ q ~ r(2) s p, ~ p ~ s(3) r s, ~ s r(4) r t, r t
Dept. Mathematics, Kyung Hee Univ.
FallaciesFallacy
An error in reasoning that results in an invalid argument.
Common fallaciesVague or ambiguous premisesBegging the question Jumping to a conclusion without adequate grounds
Converse errorp q, q p
Inverse errorp q, ~ p ~ q
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Dept. Mathematics, Kyung Hee Univ.
FallaciesA valid argument with a false conclusion
If John Lennon was a rock star, then John Lennon had red hair.John Lennon was a rock star. John Lennon had red hair.
An invalid argument with a true conclusionIf New York is a big city, then New York has tall buildings.New York has tall buildings. New York is a big city.
Dept. Mathematics, Kyung Hee Univ.
Contradictions & Valid ArgumentsContradiction rule
~ p c, where c is a contradiction p
Show that the following argument form is valid.
FFFTFTTFFTp~ p cc~ pp
conclusionpremises
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Knights & KnavesA says B is a knight.B says A and I are of opposite type. What are A and B?
Knights always tell the truth.Knaves always lie.
Dept. Mathematics, Kyung Hee Univ.
ProblemsExercise 1.311, 13, 28, 30, 33, 34, 37, 40, 44