matematika ekonomi & bisnis_1

8/19/2019 Matematika Ekonomi & Bisnis_1

1/15

MATEMATIKA EKONOMI &BISNIS

KULIAH 1

Himpunan & SistemBilangan

3/28/161 Budi Zulfachri, M.Si


2/15

Himpunan

3/28/162 Budi Zulfachri, M.Si

HIMPUNAN

PengertianHimpunan

PenyajianHimpunan

Himpunan

Universaldan

HimpunanKosong

perasionalHimpunan !"a#ungan$

Irisan$

Selisi% danPelengap

Kaida% 'Kaida%

Matematia(alamPengoperasi

anHimpunan


3/15

3/28/16Budi Zulfachri, M.Si3

Pengertian Himpunan

H i ! u

" a " • A d a l a h # u a $ u % u ! u l a " a $ a u & u & u # a " d a r i # ' ( u l a h ) * ( ' %

O * ( ' %

• O * ( ' % + ) * ( ' % a " & ' " & i # i a $ a u ' * ' " $ u % # ' * u a h h i ! u " a " d i # ' * u $ - a " & & ) $ a , a $ a u ' l ' ' " + ' l ' ' " a $ a u u " # u r .

• O * ( ' % + ) * ( ' % # u a $ u h i ! u " a " # a " & a $

* ' r a r i a # i / * i # a ) r a " & + ) r a " & $ ' r $ ' " $ u , h ' 0 a " + h ' 0 a " $ ' r $ ' " $ u , $ a " a a " + $ a " a a " $ ' r $ ' " $ u , * ' " d a + * ' " d a $ ' r $ ' " $ u , * u % u + * u % u $ ' r $ ' " $ u , a " & % a + a " & % a $ ' r $ ' " $ u d a " # ' * a & a i " a .

1 ' " a ( i a "

• 2 a l a ! ' " a ( i a " # ' c a r a u u h i ! u " a " d i l a * a " & % a " d ' " & a " h u r u f + h u r u f * ' # a r # ' ! ' r $ i , A , B , C , P , Q , R , X , Y , a $ a u Z

• S ' d a " & % a " a " & ' " ( a d i a " & & ) $ a # u a $ u h i ! u " a " d i l a * a " & % a " d ' " & a " h u r u f % ' c i l

# ' ! ' r $ i - a , b , c , p , q , r , x , y , a $ a u z


4/15

3/28/16Budi Zulfachri, M.Si

• p 4 A * ' r a r $ i * a h 0 a ) * ( ' % p

a d a l a h ' r u ! a % a " a " & & ) $ a 5 u " # u r , a $ a u ' l ' ' " 6 d a r i h i ! u " a " A

'"uli#a"a$'a$i# -

• 7 i % a # ' $ i a ! a " & & ) $ a d a r i h i ! u " a " A ( u & a ' r u ! a % a " a " & & ) $ a d a r i h i ! u " a " l a i " B , d ' " & a " ! ' r % a $ a a " l a i " p 4 A ( u & a p 4 B , a % a A d i # ' * u $ h i ! u " a " * a & i a " 5 # u * # ' $ 6 d a r i B

A B *'rar$i*ah0a A

'ru!a%a"

hi!u"a" *aia"dari B

•

2 u a * u a h h i ! u " a " d i % a $ a % a "

# a a a $ a u # ' d ' r a ( a $ a ! a * i l a # ' u a a " & & ) $ a d a r i h i ! u " a " a " & # a $ u ( u & a ' r u ! a % a " a " & & ) $ a + a " & & ) $ a * a & i h i ! u " a " a " & l a i " , d ' " & a " ! ' r % a $ a a " l a i " ( u l a h d a " ( ' " i # a " & & ) $ a + a " & & ) $ a % ' d u a h i ! u " a " $ ' r # ' * u $ # a a

A 9 B *'rar$i*ah0a hi!u"a"

A #aa d'"a"hi!u"a" B,a%"i (i%a da"

ha"a (i%a A B#'r$a B A

Pengertian Himpunan


5/15

3/28/16Budi Zulfachri, M.Si:

'"a(ia" #'*uah hi!u"a" da!a$ di$uli#%a" d'"a"dua aca cara, cara daftar da" cara kaidah.Cara daftar ialah d'"a" '"ca"$u%a" #'luruh)*('% a" '"(adi a")$a #ua$u hi!u"a". S'*aaic)"$)h-

A 9 ;1, 2, 3, , :<*'rar$i hi!u"a" A *'ra")$a%a" *ila"a"+*ila"a"*ula$ !)#i$if 1,2,3,,da" :.

Ada!u" cara kaidah ialah d'"a" '"'*u$%a"%ara%$'ri#$i% $'r$'"$u dari )*('%+)*('% a" '"(adihi!u"a" $'r#'*u$. S'*aai c)"$)h-

A 9 ; x => x >6<*'rar$i hi!u"a" A *'ra")$a%a" )*('% x , dia"a x adalah *ila"a"+*ila"a" *ula$ !)#i$if a" l'*ih*'#ar dari ")l $'$a!i l'*ih %'cil dari '"a.

?"$u% hi!u"a" A dia$a#, !'"a(ia""a #'cara %aidahda a$ di$uli#%a" #'*a ai *'ri%u$ - A 9 x 1@ x @ :

Penyajian Himpunan


6/15

A ?, B ?, da" ?,A 9 , A B, da"B ,

4 A da" (ua 4 , a%a A da" A C B, da" dilai" !iha% D C A #'r$a DC

A, B, da" ? A ?, B da" 3/28/16Budi Zulfachri, M.Si6

Hi!ua" ?"i'r#al adalah hi!ua"a a" *'ri#i%a"#'ua 'l''"+'l''" a" #'#uai u"$u% #ua$ua#alah $'r$'"$u. Bia#a"a dila*a"%a d'"a" U.

Hi!u"a" %)#)" adalah hi!u"a" a" $ida%*'ri#i%a" 'l''" #a$u !u". Hi!u"a" %)#)"dila*a"%a" d'"a" a$au F G

Mi#al%a" %i$a 'ili%i da$a *'*'ra!a hi!u"a"#'*aai *'ri%u$-

U 9 ;=,1,2,3,,:,6,,8,< A 9 ;=,1,2,3,<B

9 ;:,6,,8,


7/15


perasi Himpunan ) "a#ungan$ Irisan$ Selisi% dan Komplemen

"a#ungan 5union dari hi!u"a" A da"hi!u"a" B, di$uli#%a" d'"a" ")$a#i A U B,

adalah hi!u"a" a" *'ra")$a%a" )*('%+)*('% ili% A a$au )*('%+)*('% ili% B. A U B 9 ; x x Є A a$au x Є B}

Irisan 5intersection dari hi!u"a" A da"hi!u"a" B, di$uli#%a" d'"a" ")$a#i A ∩ B,adalah hi!u"a" a" *'ra")$a%a" *ai%)*('% ili% A au!u" )*('% ili% B d'"a"!'r%a$aa" lai", *'ra")$a%a" )*('%+)*('% a"diili%i )l'h A da" B #'cara *'r#aa.

A J B = { x; x Є A da" x Є B}

A B?

A B

?

ala hal A J B 9 , a%"i (i%aA da" B $ida% '!u"ai#a$u!u" a")$a a" diili%i*'r#aa, a%a A da" B


8/15



Selisi% dari hi!u"a" A da" hi!u"a" B,di$uli#%a" d'"a" ")$a#i A − B a$au AB, adalah

hi!u"a" a" *'ra")$a%a" )*('%+)*('% ili%A a" *u%a" )*('% ili% B. A − B ≡ AB 9 ; x x Є A $'$a!i x ¢ B}

Komplemen 5compement dari hi!u"a" A,di$uli#%a" d'"a" ")$a#i ! a$au AC a$au A" , adalahhi!u"a" a" *'ra")$a%a" )*('%+)*('% $ida%diili% )l'h A d'"a" !'r%a$aa" lai", A adalah#aa d'"a" #'li#ih a"$ara hi!u"a" u"i'r#alU da" hi!u"a" A. A = { x; x Є U $'$a!i x ¢ A} = U – A

A B?

A?


9/15


Mi#al%a" %i$a 'ili%i da$a *'*'ra!ahi!u"a" #'*aai *'ri%u$-

U 9 ;1,2,3,,:,6,,8,< P 9 ;1,2,3,,:< Q 9 ;,:,6,,8<

R 9 ;6,,8,<Ma%a -


P U Q 9;1,2,3,,:,6,,8,<P U R #$1,2,3,,:,6,,8,


10/15

3/28/16Budi Zulfachri, M.Si1=

Kaida%*aida% Matematia dalam PengoperasianHimpunan

1a. A U A = A

Kaida%

Idempoten 1*. A ∩ A = A

2a. 5 A U B) U C = A U (B U C) Kaida%Asosiati+

2*. 5 A ∩ B) J C = A J (B J C)

3a. A U B = B U AKaida%

Komutat

i+

3*. A J B = B J A

a. A U (B J C) = 5 A U B) J (B U C)

Kaida%(istri#u

ti+

*. A J (B U C) = 5 A J B) U (B J C)

:a. A U = A6a. A U U = U

Kaida%Identita

s

:*. A J = 6*. A J U = A

a. A U Ac # U8a. .AC /C 9 A

Kaida%Keleng

apan

*. A J Ac # 8*. UC 9 , C 9 U

a. .A U B/C 9 AC ∩ BCKaida%

(e

Morgan

*. .A J B/C 9 AC U BC


11/15


Lam#ang*lam#ang dalam ,eori Himpunan danArtinya

No

-

Lam#a

ng

Arti .onto% Penggunaan

1. Є A"))$a 5eemen X 4 A - )*'% L adalah a")$adari hi!u"a" A

2. Hi!u"a" *aia"5subset

A B - A adalah hi!u"a"*aia" dari B

3. U ua*u"a" 5union/ A U B - a*u"a" a"$ara A da" B. ∩ Iri#a" 50ntersection A J B - iri#a" a"$ara A da" B:. S'li#ih A B - S'li#ih a"$ara A di%ura"i

B

6. A

P K)!l''" A 9 *ila"a" !)#i$if AC 9 *ila"a" "'a$if . ? PP Hi!u"a"

?"i'r#al8. PPP Hi!u"a" K)#)"

P S'ri" (ua di$uli#%a" d'"a" ")$a#i AQPP ada !ula a" 'la*a"%a" d'"a"

1PPP #'ri" u a di$uli#%a" d'" a"


12/15


1. a*ar%a" #'*uah diara '"" u"$u% '"u"(u%%a" hi!u"a"u"i'r#al U da" hi!u"a"+hi!u"a" *aia" A #'r$a B (i%a -U # ;1,2,3,,:,6,,8<

A # ;2,3,:,<B # ;1,3,,,8<K'udia" #'l'#ai%a" -5a A - B 5c A J B 5' A J B 5* B - A 5d A U B 5f B J A

2. a*ar%a" #'*uah diara '"" a" '"u"(u%%a" hi!u"a"u"i'r#al U #'r$a hi!u"a"+hi!u"a" *aaia" A da" B u"$u% -U # ; x 3> x >1<

A # ;6,,,1=,13<

B # ;,:,11


13/15


3. A"dai%a" U # $2,3,4,%,&,',(,),*,25,22,23,24,2%+ A # $2,3,4,&,'<

B # $4,%,',(,24+C # $&,',(,),*,25,24+

a*ar%a" diara '""+"a %'udia"#'l'#ai%a" -

5a A J B 5d A U B 5 5 A U B/ J C5* B J C 5' A U B U C 5h A J B J C5c C J A 5f A J B J C 5i A J B J

?"$u% #)al "))r #a!ai d'"a" "))r 8 *'ri%u$,

a"dai%a" hi!u"a" u"i'r#al U #$5,2,3,4,%,&,',(,),*+ #'da"%a"P # $3,%,',)< da" Q # $5,&,*< #'r$a R # $4,(,*


14/15


. 5a A 5c RA 5' A

:. 5a J Q5d J R5 R J R5* ? Q 5' ? R 5h R ? R

6. 5a Q 5d J RA 5 5Q

5*R 5' A

J R 5h 5 Q/

. 5a ? 5Q J 5c 5 ? R J 5 ? 5* J 5Q ? 5d 5 J R ? 5 J

8. 5a A ? RA 5c A J RA5* 5 ? RA 5d 5 J RA


15/15

3/28/16Budi Zulfachri, M.Si1:

. B'rda#ar%a" hu%u+hu%u a$'a$i%a dala!'")!'ra#ia" hi!u"a", #'*aaia"a$'rca"$u !ada daf$ar di u%a, #'d'rha"a%a"lah!'r"a$aa"+!'r"a$aa" *'ri%u$ -5a B ? 5B ? A 5* A ? 5A J B

1=. A!a*ila U adalah #'*uah hi!u"a" u"i'r#al,

$'"$u%a" a"a a" *'"ar da" a" #alah di a"$ara!'r"a$aa"+ !'r"a$aa" di *a0ah i"i -5a A U A = U 5' C U 9 5i 5B / 9 U5* A J A = A 5f J C # 5( 5 A - C/ U U # A - C

5c B J A # B 5 6 J 9 5% B J .B - 6 / # B UU5d B U U # U 5h 6 J 6 # 6 5l 5 A U 6/ - 6 # A - 6

matematika ekonomi & bisnis_1

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