mat disk 2014 bab 1.4 + 1.5.pptx

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx

http://slidepdf.com/reader/full/mat-disk-2014-bab-14-15pptx 1/65

bilqis 1

Bab 1.4 + 1.5

Predicates & Quantifiers

Nested quantifiers

2014

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


Tujuan

• Dari bahasa anusia diterjeah!an !e

sib"l ateatis

• Dari sib"l ateatis diterjeah!an !e

bahasa anusia

bilqis 2

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis #

PREDICATE & QUANTIFIER

SUB-BAB 1.4

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 4

Predi!at dan $un%si Pr""sisi'

P(x) : x > 5Perhati!an ern(ataan P(x)) n"tasi sib"li! dari x > 5

•

X disebut subje! • > 5 disebut redi!at

• P*, belu eun(ai nilai !ebenaran selaa belu

di!etahui- disebut fun%si r""sisi P untu!

• P*, a!an enjadi r""sisi ji!a !eada telah

diberi!an nilai tertentu !eadan(a

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 5

P(x) : x > 5

• !an enjadi r""sisi ji!a telah diberi nilai

tertentu * telah dii!at / bound den%an nilai tertentu,

• Nilai (an% diberi!an !eada diabil darihiunan nilai (an% disebut seesta *universe of

discourse, atau domain

• Dala c"nt"h di atas domain daat berua

hiunan bilan%an bulat / integer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis

Pr""si"nal $uncti"n• Pr""si"nal $uncti"n '

– ebuah !andidat r""sisi

– al. 3% bl jd r"s) !rn terdiri dr satu "r lebih

ar

– N"tas P*,) P*)(, . . . . . .

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis

6 ' Pr""si"nal $uncti"n

• 6 '

– P*, 7 2 8 adalah bil %ena) 7 inte%er

– B% nilai !ebenarann(a 9

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis :

6 ' Pr""si"nal $uncti"n

• 6 '

– P*, 7 2 8 adalah bil %ena) 7 inte%er

– B% nilai !ebenarann(a 9

– ;<b '

• D"ain 8 7 inte%er ) 7 =1) 2) # . . . .>

• ?"ba diasu!!an seua nilai 8 !e dala P*,

•Tern(ata benar)

• @! nilai !ebenarann(a adalah true

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis A

?ara erubah P.$ enjadi

r""sisi• ?ara erubah P.$ enjadi r""sisi '

– Tia ar diberi nilai) atau

– @enjadi!an P.$ enjadi Quantifier

• 6' eberi nilai ada ar

– P*, 7 #) ba%aiana nilai !ebenarann(a

untu! P*4, dan P*2, 9

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 10


r""sisi• ?ara erubah P.$ enjadi r""sisi '

– Tia ar diberi nilai) atau

– @enjadi!an P.$ enjadi Quantifier

• 6' eberi nilai ada ar

– P*, 7 #) ba%aiana nilai !ebenarann(a

untu! P*4, dan P*2, 9 – ;<b '

• P*4, 7 4# nilai !ebenarann(a ' true

• P*2, 7 2# nilai !ebenarann(a ' false

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 11


r""sisi *lanjutan..,• 6' eberi nilai ada ar

– Q*)(, 7 ( + #) ba%aiana nilai

!ebenarann(a untu! Q*1)2, dan Q*#)0, 9

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 12


r""sisi *lanjutan..,• 6' eberi nilai ada ar

– Q*)(, 7 ( + #) ba%aiana nilai

!ebenarann(a untu! Q*1)2, dan Q*#)0, 9 – ;<b '

• Q*1)2, 1 7 2 + # nilai !ebenarann(a ' false

• Q*#)0, # 7 0 + # nilai !ebenarann(a ' true

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 1#

Quant!"#

• elain en%i!at den%an suatu nilai dari domain

tertentu) daat ju%a dii!at den%an quantifier

• Pr"sesn(a disebut quantification

• da # aca quantifiers'

1. Cniersal quantifier

2. 6istential quantifier

#. Cnique quantifier

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 14

Quantifiers•

Quantifiers – Cniersal quantifier ∀

– 6istential quantifier ∃

• Cniersal quantifier ∀

– Artinya :• For all

• For every

–

Ex:∀

x P (x) artinya :• Untuk semua nilai x membuat p(x) menjadibenar

• Untuk setiap nilai x membuat p(x) menjadibenar

• P(x) benar untuk semua nilai x

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 15

Cniersal quantifier ∀

• 6'

– P*n, 7 n2 + 2 n adalah bil %anjil inte%er) ba%aiana

nilai !ebenaran dari ∀ n P(n), dimana

n={inteer!

–

"#b :• P(n) is $alse karena jika n enap, maka %asilnya

jua akan enap & ex' = ($alse)& n = * (true)

• Ex: P(x) = x + - x , bm nilai kebenaran dari

∀ x P(x), dimana x=bil inteer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 1

Cniersal quantifier ∀

• 6'

– P*n, 7 n2 + 2 n adalah bil %anjil inte%er) ba%aiana

nilai !ebenaran dari ∀ n P(n), dimana

n={inteer!

–

"#b :• P(n) is $alse karena jika n enap, maka %asilnya

jua akan enap & ex' = ($alse)& n = * (true)

• Ex: P(x) = x + - x , bm nilai kebenaran dari

∀ x P(x), dimana x=bil inteer – "#b :

• Apaka% benar untuk semua x membuat P(x)

menjadi benar.

• /ernyata benar, maka ∀ x P(x) bernilai true

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 1

?"untereale

• ?"untereale

– Nilai (% eb P*, enj salah

•

6' – P*n, 7 n2 + 2 n adalah bil %anjil inte%er ) b% nilai

!ebenaran dari ∀ n P(n), dimana n=inteer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 1:

?"untereale

• ?"untereale

– Nilai (% eb P*, enj salah

•

6' – P*n, 7 n2 + 2 n adalah bil %anjil inte%er ) b% nilai

!ebenaran dari ∀ n P(n), dimana n=inteer

– "#b :

•

= p(n)=0, ternyata pernyataan ini sala%,• 1k n= adala% 2ounterexample untuk P(n)

• 1k ∀ n P(n) bernilai sala%

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 1A

UNI$ERSA% QUANTIFIER

P(x) : x > 5

x P(x) dibahasa!an dei!ian'

Euntu! seua nilai dala domain) 5F

Euntu! seua nilai ebuat P*, enjadi benarF

EP*, benar untu! seua nilai F

a!ah nilai !ebenaran dari ∀ P*, ji!a domain adalah

1. = 1) 2) #) 4) 5 > alah2. = 1..10 > alah

#. = G #) 2) 1) 0) 1) 2) #) G > alah

4. = >

Benar

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 20

UNI$ERSA% QUANTIFIER

• 6 ' Q*, 7 H 2) ba%aiana nilai !ebenaran

quantifier ∀ Q*,) diana d"ain *, adalah real

nuber 9• ;a<ab '

– 8 7 = GG2)#)4)5GGG,

– Q*, tida! benar untu! setia real nuber

– @isal Q*#, false)

– a!a 7# adalah c"untereale untu! ∀ Q*,

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 21

6istential quantifier ∃

•

6istential quantifier ∃ – Artinya :

• For some

• 1inimal ada satu nilai yan membuat menjadibenar

– Ex: ∃ x P (x) artinya :

• 1inimal ada satu nilai x membuat p(x) menjadibenar

• Ex: P(x) = x + - x , bm nilai kebenaran dari ∃ x P(x),

dimana x=bil inteer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 22


•

6istential quantifier ∃ – Artinya :

• For some

• 1inimal ada satu nilai yan membuat menjadibenar

– Ex: ∃ x P (x) artinya :

• 1inimal ada satu nilai x membuat p(x) menjadibenar

• Ex: P(x) = x + - x , bm nilai kebenaran dari ∃ x P(x),

dimana x=bil inteer – "#b :

• Apaka% benar minimal ada satu nilai x membuatP(x) menjadi benar.

•

/ernyata benar, maka ∃ x P(x) bernilai true

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 2#


•

Ex: P(x) = x - 3 , bm nilai kebenaran dari ∃ x P(x),dimana x=bil inteer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 24


•

Ex: P(x) = x - 3 , bm nilai kebenaran dari ∃ x P(x),dimana x=bil inteer

– "#b :


• /ernyata benar, maka ∃ x P(x) bernilai true

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 25


•

Ex: P(x) x = x + , bm nilai kebenaran dari∃ xP(x), dimana x=bil inteer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 2


•

Ex: P(x) x = x + , bm nilai kebenaran dari∃ xP(x), dimana x=bil inteer

– "#b :


• /ernyata tidak ada satu nilaipun untuk x yanbenar, maka ∃ x P(x) bernilai $alse

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 2

EXISTENTIA% QUANTIFIER

P(x) : x > 5

x P(x) dibahasa!an dei!ian'

Einial ada satu nilai (an% ebuat P*, enjadi benarF

a!ah nilai !ebenaran dari P*, ji!a domain adalah

1. = 1) 2) #) 4) 5 > salah

2. = 1..10 > benar #. = G #) 2) 1) 0) 1) 2) #) G > benar

4. = > benar

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 2:

UNIQUE QUANTIFIER

P(x) : x > 5

! x P(x) dibahasa!an dei!ian'

Ehan(a ada 1 nilai (an% ebuat P*, enjadi benarF

a!ah nilai !ebenaran dari ! P*, ji!a domain adalah

1. = 1) 2) #) 4) 5) > benar

2. = 1..10 > salah#. = G #) 2) 1) 0) 1) 2) #) G > salah

4. = > benar

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 2A

$ARIABE% TERIAT' $ARIABE% BEBAS' SCOPE

?"nt"h 1' P(x) : x > 5

dala r""sisi P*4,) ∀ P*,) ∃ P*,

ariabel disebut ariabel teri!at

P(4) ' dii!at "leh nilai 4

x P(x) ' dii!at "leh x

x P(x) ' dii!at "leh x

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis #0


?"nt"h 2' P(x' ) : x > 5

P(4' ) adalah ariabel teri!at *"leh 4,( disebut ariabel bebas

P(x' ) disebut ariabel bebas ( dii!at "leh ∀(

x P(x' ) dii!at "leh ∃( disebut ariabel bebas

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis #1


?"nt"h #' x * P(x)∨

Q(x) +∧

x R(x)

scope dari x adalah I P*,∨

Q*, J

scope dari x adalah K*,

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis #2

NE,ASI

Cnt:1. eua ahasis<a di !elas ini lulus dala ata!uliah

$isi!a.

2. Tida! seua ahasis<a di !elas ini lulus dala

ata!uliah $isi!a.

D"ain *, 7 = ahasis<a di !elas ini >

$*, ' lulus dala ata!uliah $isi!a

1. ∀ $*,2. ¬∀ $*, e!ialen den%an ∃ I¬ $*,J

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis ##

NE,ASI

Cnt:

1. da ahasis<a dari @ala(sia di !elas ini.

2. Tida! ada ahasis<a dari @ala(sia di !elas ini.

D"ain *, 7 = ahasis<a di !elas ini >

@*, ' dari @ala(sia

1. ∃ @*,

2. ¬ ∃ @*, e!ialen den%an ∀ I¬ @*,J

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 20 hal 4

• What are the negations of thestatements

– “There is an honest politician” and – “All Americans eat cheeseburgers”?

bilqis #4

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<ab

• D"ain *, 7 "litician

• L *, 7 is h"nest

• ∃ 8 L *, 7 there is an h"nest "litician

• Ne%atif

• M ∃ 8 L *, e!ialen ∀ I¬ L*,J

• 7 6er( "litician is dish"nest

bilqis #5

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<ab

• D"ain *, 7 aericans

• ? *, 7 eat bur%ers

• ∀ 8 ? *, 7 ll aericans eat bur%ers

• Ne%atif

• M ∀ 8 ? *, e!ialen ∃ I¬ ?*,J

• 7 there is aericans d"es n"t eat bur%ers

bilqis #

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis #

NE,ASI

Ne%asi Pr""sisie!ialen

TKC6 $6*ne%asi,

¬∃ P*,

¬∀

P*,

∀ *¬

P*,,

∃

*¬

P*,,

untu! seua )P*, salah

ada suatu nilai (an% ebuatP*, salah

ada suatu nilai (an% ebuatP*, benar

untu! seua )P*, benar

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis #:

r""sisi TKC6 $6 *ne%asi,

∀ ∀( P*) (,

∀( ∀ P*) (,

P*) (, benar untu! seua asan%an dan (

da asan%an dan ( (an%ebuat P*) (, salah

∀ ∃( P*) (, Cntu! tia ada suatu ( (an%ebuat P*) (, benar

da (an% ebuatP*) (, salah untu! tia (

∃ ∀( P*) (, da (an% ebuat P*) (, benar untu! tia (

Cntu! tia ada suatu ((an% ebuat P*) (, salah

∃ ∃( P*) (,

∃ ∃( P*) (,

da asan%an dan ( (an%ebuat P*) (, benar

P*) (, salah untu! seua asan%an dan (

Kin%!asan untu! 2 quantifiers

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6 ' 2# hal 4:

• 6ress the stateent E6er( student in this

class has studied calculusF usin% redicates

and quantifiers.

bilqis #A

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<ab

• 8 7 student in this class subject

• ? * , 7 studied calculus

• ∀ ? *, 7 eer( student in this class has

studied calculus

bilqis 40

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<aban lain

• 8 7 student

• *, 7 in this class

• ? * , 7 studied calculus

• ∀ * *, ? *,, 7 eer( student) if in

this class then has studied calculus

bilqis 41

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 2 hal 50

• ?"nsider these stateents. The first t<" are

called premises and the third is called the

conclusion. The entire set is called anargument .

– Ell li"ns are fierce.F

–

E"e li"ns d" n"t drin! c"ffee.F – E"e fierce creatures d" n"t drin! c"ffee.F

bilqis 42

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<ab

• D"ain *, 7 anial

• P *, 7 is li"n

• Q *, 7 is fierce

• K *, 7 drin!s c"ffee

• Oe can eress these stateents as'

– ∀ x(P(x) Q(x)).

– ∃ x(P(x) ∧ ￢ R(x)).

– ∃ x(Q(x) ∧ ￢ R(x)).

bilqis 4#

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 2 hal 51

• ?"nsider these stateents) "f <hich the first

three are reises and the f"urth is a alid

c"nclusi"n. – Ell huin%birds are richl( c"l"red.F

– EN" lar%e birds lie "n h"ne(.F

–

EBirds that d" n"t lie "n h"ne( are dull inc"l"r.F

– ELuin%birds are sall.F

bilqis 44

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<ab

• D"ain *, 7 all creature

• P *, 7 is li"n

• Q *, 7 is fierce

• K *, 7 drin!s c"ffee

• Oe can eress the stateents in the ar%uent as

• ∀ x(P(x) S(x)).

• ￢∃ x(Q(x) ∧ R(x)).

• ∀ x( ￢ R(x)￢ S(x)).

• ∀ x(P(x)￢ Q(x)).

– all 7 n"t lar%e

–Dull in c"l"r 7 n"t richl( c"l"red bilqis 45

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 4

1.5 N"/t"0 Quant!"#/

PRPSISI DEN,AN

BEBERAPA QUANTIFIER

n%%a domain adalah hiunan bilan%an n(ata*real numbers,

1.

x * + ( 7 ( +,

2. x * + ( 7 0,

3.

x * + ( 7 0,

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 4

D"uble ariable•

6' P*)(, +( 7 (+ – B% nilai !ebenaran untu! ∀ x ∀ y P(x,y),

dimana x,y = inteer .

– "#b :• Cntu! seua dan untu! seua () aa!ah ebuat

P*)(, benar 9 a!ah +(7(+ adalah benar 9

• Tern(ata benar) a!a r""sisi ini berniali benar

•

6' Q*)(, +( 7 0 – B% nilai !ebenaran untu! '

• ∃ ( ∀ x 4(x,y)

• ∀ x ∃ ( 4(x,y)

• 5imana x,y adala% inteer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 4:

• ;<b ' – ∃ ( ∀ x 4(x,y) :

• 1in ada nilai y untuk semua6setiap nilai xyan membuat 4(x,y) benar

• Ada y yan membuat 4(x,y) benar untuk tiap x

• 1is y=, mk x = 7 , pada%al se%arusnya ini

bisa untuk sembaran nilai x, maka• Proposisi ini bernilai $alse

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 4A

• ;<b ' – ∀ x ∃ ( 4(x,y)

• Untuk semua nilai x min ada nilai y ymembuat 4(x,y) benar

• Untuk tiap nilai x ada satu y yan membuat4(x,y) benar

• 1is x=, mk y=7• 1is x=3, mk y=73

• 1k =- proposisi ini bernilai true

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 50

Trile ariable•

6' Q*)(), +( 7 – B% nilai !ebenaran untu! '

• ∀ x ∀ y ∃ 4(x,y,8)

• ∃ x ∀ y ∀ 4(x,y,8)

• 5imana x,y,8 adala% inteer

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


bilqis 51

• ;<b ' – ∀ x ∀ y ∃ 4(x,y,8) :

• 1in ada nilai 8 untuk semua6setiap nilai x dany yan membuat 4(x,y,8) benar

• 1is y= dan x = 3 , maka 8 = *

• Proposisi ini bernilai benar

– ∃ x ∀ y ∀ 4(x,y,8)• 1in ada nilai x untuk semua nilai y dan 8 yan

membuat 4(x,y,8) benar

• 1Emba2a dari kiri ke kanan,

• "adi x = apaka% bisa untuk sembaran nilai ydan 8 , ja#aban tidak

• 1k =- proposisi ini bernilai $alse

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


@enerjeah!an bhs in%%ris

l"%ical eressi"n• E6er( student in this class has studied

calculusF

• ;a<ab – D"ain *, 7 student in this class

– ? *, 7 has studied calculus

–"%ical 6ressi"n *6, ∀ x 9 (x)

bilqis 52

• Ell li"ns are Rerce.F

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


ll li"ns are Rerce.

• E"e li"ns d" n"t drin! c"ffee.F

• E"e Rerce creatures d" n"t drin! c"ffee.F

• ;a<ab '

– D"ain *, 7 anial

– P*, 7 is a li"n

– Q*, 7 is fierce

– K *, 7 drin!s c"ffee

– 6

• ∀ x ( p(x) (x)• ∃ * *, S M r*8,, tida! b"leh ∃ **8, Mr*8,) !arena

a!an true ji!a bu!an li"n) seharusn(a inial ada 1 li"n

• ∃ * q*, S M K*8,,

bilqis 5#

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6.

Translate the stateent

EThe su "f t<" "sitie inte%ers is al<a(s

"sitieFint" a l"%ical eressi"n.

bilqis 54

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<aban ertaa

• E$"r eer( t<" inte%ers) if these inte%ers are b"th "sitie) then the su "f

these inte%ers is "sitie.F

• Net)<e intr"duce the ariables and ( t" "btain

• E$"r all "sitie inte%ers and () +( is "sitie.F

• ?"nsequentl()

• D"ain all inte%er

• ∀ (** 0, *( 0, * +(0,,) ∀ ∧

bilqis 55

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


;a<aban !edua

–D"ain "sitie inte%ers.

– EThe su "f t<" "sitie inte%ers is

al<a(s "sitieF bec"es – E$"r eer( t<" "sitie inte%ers) the su

"f these inte%ers is "sitie.

– ∀ (*+(0, ∀

bilqis 5

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. A

• Translate the stateent

• ∀*?*, (*?*(, $*)(,,, int" 6n%lish)∨∃ ∧

• Ohere – d"ain f"r b"th and ( all students in ("ur sch""l.

– ?*,is E has a c"uter)F

– $*)(,is E and ( are friends)F and the

• "luti"n'

• f"r eer( student in ("ur sch""l) has a c"uter "r

there is a student ( such that ( has a c"uter and and (are friends. atau

• eer( student in ("ur sch""l has a c"uter "r has a friend

<h" has a c"uter. bilqis 5

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 10• Translate the stateent

• ∃ ( **$*)(, $*), *( ,,U$*(),,∀ ∀ ∧ ∧ en%lish• <here

– d"ain f"r ) () and "f all students in ("ur sch""l

– $*a)b,eans a and b are friends and the.

•"luti"n'

• *$*)(, $*), *( ,,U$*(),.∧ ∧ if students and ( are

friends) and students and are friends) and further"re) if ( and

are n"t the sae student) then ( and are n"t friends.

•;a<abann(a there is a student such that f"r all students ( andall students "ther than () if and ( are friends and and are

friends) then ( and are n"t friends.

• Vn "ther <"rds) there is a student n"ne "f <h"se friends are als"

friends <ith each "ther. bilqis 5:

6 11

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 11• 6ress the stateent

• EVf a ers"n is feale and is a arent) then this ers"n is s"e"neWs

"therF.• "luti"n'

• aa artin(a den%an $"r eer( ers"n ) if ers"n is feale and

ers"n is a arent) then there eists a ers"n ( such that ers"n is

the "ther "f ers"n (.F

• r""siti"nal functi"ns – D"ain 7 ers"n

– $*,E is feale)F

– P*, E is a arent)F and

– @*)(, E is the "ther "f (.F

• ∀**$*, P*,, (@*)(,,.∧ ∃

• <e can "e ( t" the left) because ( d"es n"t aear in∃

$*, P*,.∧

•

Lasil a!hir (**$*, P*,, @*)(,,.∀ ∃ ∧ bilqis 5A

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 12

• 6ress the stateent E6er("ne has eactl( "ne best friendF as a

l"%ical eressi"n in"lin% redicates) quantiRers <ith a d"ain

c"nsistin% "f all e"le) and l"%ical c"nnecties.

• "luti"n' The stateent E6er("ne has eactl( "ne best friendF can be

eressed as E$"r eer( ers"n ) ers"n has eactl( "ne bestfriend.F Vntr"ducin% the uniersal quantiRer) <e see that this stateent

is the sae as E *ers"n has eactl( "ne best friend,)F <here the ∀

d"ain c"nsists "f all e"le. T" sa( that has eactl( "ne best friend

eans that there is a ers"n ( <h" is the best friend "f ) and

further"re) that f"r eer( ers"n ) if ers"n is n"t ers"n () then is n"t the best friend "f . Ohen <e intr"duce the redicate B*)(, t"

be the stateent E( is the best friend "f )F the stateent that has

eactl( "ne best friend can be reresented as (*B*)(, ** 7∃ ∧∀

(,UB*),,,. ?"nsequentl() "ur "ri%inal stateent can be eressed

as (*B*)(, ** 7 (,UB*),,, ∀ ∃ ∧∀ bilqis 0

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 1#

• Cse quantiRers t" eress the stateent EThere is a

<"an <h" has ta!en a Xi%ht "n eer( airline in the

<"rld.F

• "luti"n' et P*<)f,be E< has ta!en fF and Q*f)a, be Ef is a

Xi%ht "n a.F Oe can eress the stateent as

< a f*P*<)f, Q*f)a,,)∃ ∀ ∃ ∧

<herethed"ains"fdisc"ursef"r<)f)anda

c"nsist"fallthe<"eninthe<"rld)allairlane Xi%hts) and all

airlines) resectiel(. The stateent c"uld als" beeressed as < a fK*<)f)a,) <here K*<)f)a,is E< has∃ ∀ ∃

ta!en f "n a.Flth"u%h this is "re c"act) it s"e<hat

"bscures the relati"nshis a"n% the ariables.

?"nsequentl() the Rrst s"luti"n is usuall( referable. Y bilqis 1

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 14• 6ress the ne%ati"n "f the stateent (*( 7 1,∀ ∃

s" that n" ne%ati"n recedes a quantiRer. "luti"n'

B( successiel( al(in% De @"r%anWs la<s f"r

quantiRers in Table 2 "f ecti"n 1.4) <e can "ethe ne%ati"n in U (*( 7 1, inside all the∀ ∃

quantiRers. Oe Rnd thatU (*( 7 1, is∀ ∃

equialent t" U (*( 7 1,) <hich is equialent∃ ∃

t" (U*( 7 1,. Because U*( 7 1, can be∃ ∀eressed "re sil( as ( 7 1) <e c"nclude that

"ur ne%ated stateent can be eressed as (*(∃ ∀

7 1,. bilqis 2

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


6. 15• Cse quantiRers t" eress the stateent that EThere d"es n"t eist a

<"an <h" has ta!en a Xi%ht "n eer( airline in the <"rld.F

• "luti"n' This stateent is the ne%ati"n "f the stateent EThere is a

<"an <h" has ta!en a Xi%ht "n eer( airline in the <"rldF fr"

6ale 1#. B( 6ale 1#) "ur stateent can be eressed asU < a f*P*<)f, Q*f)a,,) <here P*<)f,is E< has ta!en fF and Q*f)a, is∃ ∀ ∃ ∧

Ef is a Xi%ht "n a.F B( successiel( al(in% De @"r%anWs la<s f"r

quantiRers in Table 2 "f ecti"n 1.4 t" "e the ne%ati"n inside

successie quantiRers and b( al(in% De @"r%anWs la< f"r ne%atin% a

c"njuncti"n in the last ste) <e Rnd that "ur stateent is equialent t"

each "f this sequence "f stateents' <U a f*P*<)f, Q*f)a,, ∀ ∀ ∃ ∧

Z < aU f*P*<)f, Q*f)a,, Z < a fU*P*<)f, Q*f)a,, ∀ ∃ ∃ ∧ ∀ ∃ ∀ ∧

Z < a f*UP*<)f, UQ*f)a,,. This last stateent states E$"r eer( ∀ ∃ ∀ ∨

<"an there is an airline such that f"r all Xi%hts) this <"an has n"t

ta!en that Xi%ht "r that Xi%ht is n"t "n this airline.F bilqis #

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


• Tu%as el""!

– cari 2 s"al dan ja<aban di internet (an%

berhubun%an den%an ateri t ini – Tulis alaat internetn(a

– Di !iri !e bilqis[itssb(.edu) tera!hir

• @in%%u) 21 eteber 2014 ja 1.00

• $"rat subject + naa file

– atdis!?tu%as\1a<ar

– Bentu! t inf"rasi naa !el""! +

an%%"ta bilqis 4

mailto:[email protected]

mailto:[email protected]

7/16/2019 mat disk 2014 bab 1.4 + 1.5.pptx


PK

• ?ari di internet catat alaatn(a

• 1 c"nt"h s"al ' ada beberaa !aliat

*inial #, (an% salin% berhubun%an danterjeah!an !e dala quantifier

• Bentu! file PPT + naa !el""!

• Di!uul!an di freeshare

mat disk 2014 bab 1.4 + 1.5.pptx

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