lecture note chap 01

Dept. Mathematics, Kyung Hee Univ.

Discrete Mathematicswith Applications

LEE, SOOJOONDepartment of Mathematics

Kyung Hee University


Chapter 1 The Logic of Compound Statements

1.1 Logical Form & Logical Equivalence1.2 Conditional Statements

1.3 Valid and Invalid Arguments


1.1 Logical Form and Logical Equivalence


Statement (or Proposition)A sentence that is true or false but not both.


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


Truth ValuesEither true (T) or false (F)Truth table

TFFT

~ pp

FFFFTFFFTTTT

p qqp

FFFTTFTFTTTT

p qqp


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


Truth Table for Exclusive ORp q = (p q ) ~ ( p q )

FFTFFTTT

(p q ) ~ ( p q )qp


Truth Table for ( p q ) ~ r

FFFTFFFTFTTFFFTTFTFTTTTT

( p q ) ~ rrqp


Logical EquivalenceTwo statement forms are called logically equivalentif and only if they have identical truth table.Notation: PQExample: Double negation

~ (~ p) p


Nonequivalence~ ( p q ) T ~ p ~ q~ ( p q ) T ~ p ~ q


De Morgans Laws~ ( p q ) ~ p ~ q~ ( p q ) ~ p ~ q


Tautologies & Contradictions Tautology

A statement form that is always true Contradiction

A statement form that is always false


p t p, p c ct : a tautology c : a contradiction

FFFFFT

p ccp

FTFTTT

p ttp


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


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


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


ProblemsExercise 1.1 8, 9, 15, 28, 30, 39


1.2 Conditional Statements


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


Truth table for p ~ q ~ p

T T TFFF T TTFT F FFTT F FTT

p ~ q ~ pqp

hypothesis conclusion


(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


p q~ p q

TTFFTTTFFFFTTTTT

~ p qp qqp


~ (p q) p ~ qThe negation of if p then q is logically equivalent to p and not q.~ (p q) ~ (~ p q) p ~ q


ContrapositiveThe contrapositiveof p q is ~ q ~ pp q~ q ~ p

p q~ p q ~ (~ q) ~ p ~ q ~ p

TTFFTTTFFFFTTTTT

~ q ~ pp qqp


Converse & InverseThe converseof p q is q pThe inverseof p q is ~ p ~ q

TTTFFFFTTFTTFFTTTTTT

~ p ~ qq pp qqp


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


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)


ProblemsExercise 1.2 8, 11, 20, 23, 40, 41


1.3 Valid and Invalid Arguments


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


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


Invalid Argument Formp q ~ rq p r p r


Valid Argument Formp (q r)~ r p q


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


Modus Ponens & Modus Tollens

FFTTp

TTTFTFFTFTTTTFFTFFFFTFFTTTTT

~ p~ qp qqp qqp


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


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?


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


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


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.


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


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.


ProblemsExercise 1.311, 13, 28, 30, 33, 34, 37, 40, 44

lecture note chap 01

Documents

p qif p

p q fftfftttp q

p p q pnegation of t

rffftffftfttfffttftfttttt

p q qpdept

p c cabsorption lawsp

p qp q r p q r

commutative lawsp q