2 Predicates and Predicated Logic

7/25/2019 2 Predicates and Predicated Logic

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PREDICATES AND

PREDICATED LOGICLecture by

Ms. Cherry Rose R.Estabillo

MATH102C C.


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MATH102C C.

Predicate Logic is a e!tesioo" Pro#ositioal Logic.

It $as used to e!#ress the%eaig o" $ide rage o"state%ets i %athe%atics

ad co%#uter sciece i$ays that #er%it us toreaso ad e!#lore

relatioshi#s bet$ee

PREDICATE LOGIC


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Consider the statement

! " #! $ % & 2

'Com()ter ! is )nder atta*+,% ha*+ers.-

MATH102C C.

neither true nor falsewhen the values of thevariables are not specied


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To *om(onents

The 'ariable / s),e*t o the statementPredicate/ reers to a (ro(ert% that thes),e*t o the statement *an hae.

MATH102C C.


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MATH102C C.

Consider3 '! " #-

aria,4e !

P is 5reater than #Pro#ositioal "uctio Pat !( P)!*

P678P618


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MATH102C C.

Consider3 '!$%&2-

aria,4e !3 %

Predi*ate 9Pro#ositioal "uctio + at!,y( +)!,y*

9613:896;3#8


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MATH102C C.

A statement can have

more than onevariable.A statement o the orm

P6!13!23


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MATH102C C.

Exercise

1.8 Let P6!8 denote thestatement '!>?.- @hat arethe tr)th a4)es

a.* P)* b* P)/*c.* P)0*

2.8 Let P6!8 denote thestatement 'the ord !

*ontains the 4etter a.- @hatare the tr)th a4)esa.* P)orage* b*


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MATH102C C.

Determine the tr)th a4)es o the .statements

618 P6!8 :! & # : P6283 P6?8

628 P6!3 %8 2! / :% $ ? P603 18 3 P63 18

6:8 P6!3 %3 F8 ! / % F P613 13183 P623=13 08

THE TRTH ALES


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MATH102C C.

aria,4e !

is an inte5er

9 is a rationa4 n)m,er

Jine is not an inte5er.

Jine is an inte5er and Fero is a rationa4

n)m,er.I nine is an inte5er3 then Fero is arationa4 n)m,er.

These propositions can be negatedor conjoined


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MATH102C C.

Let P)!* be a state%eti'ol'ig the 'ariable ! adlet D be a set. 1e call P aPROPOSITIONAL 23NCTION)$rt D* i" "or each ! i D, P)!*

is a PROPOSITION. 1e call Dthe DOMAIN O2 DISCO3RSE.

PROPOSITIOJAL KJCTIOJ


12/41

MATH102C C.

Let

P6!8 ! & 2!2is a rationa4n)m,er.

D set o rationa4 n)m,ers

P6!8 St)dent ! s*ored (ere*t

in the test.

D set o st)dents in ST


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MATH102C C.

+3ANTI2ICATION

- used to create a#ro#ositio "ro% a#ro#ositioal "uctio. It

e!#resses the e!tet to$hich a #redicate is trueo'er a rage o" ele%ets.

T1O T4PES O2+3ANTI2ICATION

)5*3NI6ERSAL


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MATH102C C.

T9E 3NI6ERSAL +3ANTI2IER

Let P6!8 ,e a (ro(osition )n*tion ithdomain o dis*o)rse D.

The statement for every x, !x" is said to ,e3NI6ERSAL +3ANTI2IED STATEMENT.

The s%m,o4 6universal quantifer8means 'or eer% or or A44-

The statement 'or a44 !3 P6!8- *an ,eritten as !P6!8.


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MATH102C C.

T9E 3NI6ERSAL +3ANTI2IER

The statement or a44 !3 P6!8 is TR3E

i P6!8 is tr)e or eer% ! in D.

The statement or a44 !3 P6!8 is 2ALSE

i P6!8 is a4se or at 4east one ! in

D.


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MATH102C C.

)5*Let P)!* be the state%et : ! ; 7 uati?catio F!P)!*, $here thedo%ai cosist o" all o-egati'eitegers

)7* 1hat is the truth 'alue o" F!P)!*,$here P)!* is the state%et :!7B =ad the do%ai cosists o" itegers

greater tha

E!am(4es


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MATH102C C.

Let P)!* be the state%et :! has

'isited the Museu%= $here thedo%ai cosists o" the studets 3ST.E!#ress each o" these

>uati?catios i Eglish.)5*!P)!*

)7*!P)!*

)*@!P)!*)/*@! P)!*

E!er*ises


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MATH102C C.

2or each state%et, ?d ado%ai "or $hich the state%et is true ad do%ai$here the "ollo$ig state%et

is "alse.

5.* E'eryoe s#eaHs Ilocao.

7.* There is so%eoe older tha75 years.

E!er*ises


22/41

MATH102C C.

Traslate these state%ets ito

Eglish, $here C)!* is :! is aco%edia= ad 2)!* is :! is "uy=ad the do%ai cosists o" all

#eo#le.)5*@!)C)!* 2)!**

)7*!)C)!* 2)!**

)*@!)C)!* 2)!**)/*!)C)!* 2)!**

E!er*ises


23/41

MATH102C C.

E!er*iseLet C6!8 ,e the statement '! has a *at-3 4et D6!8 ,e

the statement '! has a do5-3 and 4et H6!8 ,e thestatement '! has a hamster-. E!(ress ea*h o thesestatement in terms o C6!83 D6!83 H6!83 )antiers3 and4o5i*a4 *onne*ties. Let the domain *onsist o a44st)dents in %o)r *4ass.

1.8 A st)dent in %o)r *4ass has a *at3 a do5 and ahamster.

2.8 A44 st)dent in %o)r *4ass has a *at3 a do5 or ahamster.

:.8 Some st)dent in %o)r *4ass has a *at and ahamster3 ,)t not a do5.


24/41

MATH102C C.

3NI+3ENESS +3ANTI2IERThe statement 'There e!ists a

)ni)e ! s)*h that P6!8 is tr)e-

or 'there is e!a*t4% one- or'there is one and on4% one- is ane!am(4e o )anti*ation )sin5

3NI+3ENESS +3ANTI2IER.And this *an ,e ritten as J

!P)!*.

OT9ER +3ANTI2IERS


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MATH102C C.

+uati?ers $ith RestrictedDo%ai

To restri*t a domain3 an a,,reiatednotation is oten )sed

E!am(4e 1 @!B )!K


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MATH102C C.

E!am(4e 2 @y )y *

here the domain is a44rea4 n)m,ers

The cube o a nonzero real

number is nonzero.

@y)y y*


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MATH102C C.

The >uati?ers ad @ ha'ehigher #recedece tha alllogical o#eratios.

E!a%#le.

The co&uctio o" !P)!* ad+)!* ( )!P)!**+)!* rather tha!)P)!*+)!**.

PRECEDEJCE 9AJTIKIER


28/41

MATH102C C.

The o**)rren*e o the aria,4e is said to ,eBOUND hen the )antier is )sed onthe aria,4e.

The o**)rren*e o the aria,4e that is not,o)nd ,% a )antier is said to ,e F!!.

The (art o a 4o5i*a4 e!(ression to hi*h a

)antier is a((4ied *a44ed the"#O$!o)antier.

BIJDIJG ARIABLES


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MATH102C C.

)5*@!)/! ; /y B7*

oud( ! )by ui'ersal >uati?er*

2ree( y Q

)7* !)P)!*+)!**@!R)!*

oud( all 'ariables

2ree( DOES NOT E8IST

Sco#e o" !( P)!*+)!* Sco#e o" @!(R)!*

E!am(4e


30/41

MATH102C C.

JEGATIJG 9AJTIKIEDENPRESSIOJS

NEGATION E+3I6ALEN

TSTATEMENT

19EN IS

NEGATIONTR3E

19EN IS

NEGATION 2ALSE

!P)!* @!P)!* 2or e'ery

!, P)!* is"alse

There is a !

"or $hichP)!* is true.

@!P)!*

!P)!* There is a! "or $hich

P)!* is"alse.

P)!* is true"or e'ery !.

Note( The rules "or egatios "or>uati?ers are called DE MORGANS LA1S

2OR +3ANTI2IERS


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MATH102C C.

1hat are the egatios o"

5.* @!)!K

7.* !)!KU*

Sho$ that @!)P)!* +)!**ad !)P)!*+)!** arelogically e>ui'alet.

E!a%#les


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MATH102C C.

T@O=PLACE PREDICATES arereerred to as re4ationa4

(redi*ates3 the% e!(ress are4ation ,eteen to*om(onents.

Let P6!3%8 ! is easier than %!P6!3 %8 Some ! is easier than %.

%P6!3%8 ! is easier than eer% %.

T1O VPLACE PREDICATES


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MATH102C C.

T@O=PLACE PREDICATES arereerred to as re4ationa4

(redi*ates3 the% e!(ress are4ation ,eteen to*om(onents.

Let P is easier thanP6!3%8 ! is easier than %

!P6!3 %8 Some ! is easier than %.

T1O VPLACE PREDICATES


34/41

MATH102C C.

To )antiers are JESTED i one)antier is ithin the s*o(e o the other)antier.

ENAMPLE

!%66!086%0886!% 08

Consider that the domain o dis*o)rse or,oth aria,4es are rea4 n)m,ers.

'Kor a44 rea4 n)m,er ! and or a44 rea4n)m,er %3 i ! is 4ess than 0 or % 4essthan 03 then !% is 4ess than 0.-

JESTED 9AJTIKIERS


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E!er*iseLet P6!3%8 ,e the statement 'st)dent !has ta+en %-3 here domain o ! *onsistso a44 st)dents o IICS and % *onsists oa44 Math *o)rses. E!(ress ea*h o these

)anti*ations in En54ish senten*es.1.8 !%P6!3%8 ?.8 !%P6!3%8

2.8 %!P6!3%8 #.8 !%P6!3%8

:.8 !%P6!3%8 .8 %!P6!3%8

THE ORDER OK


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MATH102C C.

It is im(ortant to note that theorder o the )antier3 )n4ess a44the )antiers are

)niersa46e!istentia48 )antiers.

THE ORDER OK9AJTIKIERS

THE ORDER OK


37/41

MATH102C C.

Let P6!3 %8 2!% $ :! & %@hat are the tr)th a4)es o

!%P6!3 %8 %!P6!3 %8

!%P6!3 %8 !%P6!3 %8

!%P6!3 %8 %!P6!3 %8

here the domain or a44 aria,4es*onsists o a44 rea4 n)m,ers

THE ORDER OK9AJTIKIERS

9AJTIKICATIOJ OK T@O


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MATH102C C.

9AJTIKICATIOJ OK T@OARIABLES

STATEMENT

19EN TR3E 19EN 2ALSE

@!@yP)!,y*@y@!P)!,

y*

P)!, y* is true "ore'ery #air !, y.

There is a #air !, y "or$hich P)!, y* is "alse.

@!yP)!,y*

2or e'ery !, thereis a y "or $hichP)!, y* is true.

There is a ! suchthat P)!, y* is "alse"or e'ery y.

!@yP)!,y*

There is a ! "or$hich P)!, y* istrue "or e'ery y.

2or e'ery ! there is ay "or $hich P)!, y* is"alse.

!yP)!,

y*

There is #air !, y

"or $hich P)!, y*

P)!, y* is "alse "or

e'ery !, y.


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MATH102C C.

Let P6!3 %8 2! & % $#@hat are the tr)th a4)es o

!%P6!3 %8 %!P6!3 %8

!%P6!3 %8 !%P6!3 %8

!%P6!3 %8 %!P6!3 %8

here the domain or a44 aria,4es*onsists o a44 inte5ers

E!er*ise


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MATH102C C.

1.8Let P6!3 %8 !Q & 1 " % & 1@hat are the tr)th a4)es o

a.8!%P6!3 %8 *.8 !%P6!3 %8

,.8 !%P6!3 %8 d.8 !%P6!3 %8

here the domain or a44 aria,4es *onsists oa44 inte5ers

2.8 E!(ress the ne5ations o ea*h statements

so that ne5ation s%m,o4s immediate4%(re*edes (redi*ates.

a.8 !%6P6!3%896!3%88

,.8 !%P6!3%8!%96!3%8

E!er*ise

JEGATIJG JESTED


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02C C

E!(ress the ne5ations o ea*hstatements so that ne5ations%m,o4s immediate4% (re*edes

(redi*ates.1.8 !%6P6!3%896!3%88

2.8 !%P6!3%8!%96!3%8

JEGATIJG JESTED9AJTIKIERS

2 Predicates and Predicated Logic

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