latihan soal fungsi matematika diskrit

7/24/2019 Latihan Soal Fungsi Matematika Diskrit

1/13

FUNCTION, RECURSION AND FUZZY SET

A function f from a setX to a set Y, denoted f :X Y , is a relation fromX, the domain, to Y , the codomain, that satises two properties: (1) everyelement in X is related to some element in Y , and (2) no element in X is

related to more than one element in Y .

The set of all values of f taken toether is !alled the range of f or the imageof X under f.

"#$1:A%&1,2,', %&2,',*,+,, f:A s-- f(#) % 2#ind the domain, !o$domain, and rane%imae(f). /raw the arrow$diaram.1,2,' ea!h is !alled the pre$imae and 2,*,+ ea!h is !alled the imae.0ow many fun!tions are availa-le from A to 0ow many fun!tions are availa-le from to A

"#$2:hi!h one is a fun!tion

"#$':hi!h one is a fun!tion 3tate the domain and rane.


2/13

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-2

2

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-2 2

y

-4

-2

2

4

x-2 2

y

-4

-2

2

4

A B C D

E F G H

I J K L

Onetoone (inecti!e" Function4et F -e a fun!tion from a setX to a set Y. F is onetoone (or inecti!e) if,and only if, for all elementsx1andx2 in X, if F(x1) % F(x2), thenx1% x2, or,e5uivalently, ifx1 x2, then F(x1) F(x2).3ym-oli!ally, F:X Y is one$to$one 6 7x1, x28X, if F(x1) % F(x2) thenx1%x2."#$*:

is one$to$one -ut 9 is not."#$: hi!h one is one$to$one fun!tion


3/13

Onto (Surecti!e" Function4et -e a fun!tion from a set ; to a set


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a b c d

e f g h

"#$@:hi!h one is an in=e!tive, a sur=e!tive, or a -i=e!tive fun!tion

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-2

2

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-4 -2 2 4

y

-4

-2

2

4

x-2 2

y

-4

-2

2

4

x-2 2

y

-4

-2

2

4

A B C D

E F G H

I J K L

A one$to$one (1$1) fun!tion (in=e!tive fun!tion) o!!urs if ea!hhoriontal line !uts the fun!tion at most at one point.

An onto$fun!tion (sur=e!tive fun!tion) o!!urs if rane % !o$domain, orea!h horiontal line !uts the fun!tion at least at one point.

A -i=e!tive$fun!tion (or one$to$one !orresponden!e) o!!urs if ea!h

horiontal line !uts the fun!tion at e#a!tly one point.

"#$1B:a. Cf A%&1,2,' and %&a,-,!, how many one$to$one !orresponden!e are

there from A to -. Cf n(A)%m and n() % m, how many one$to$one !orresponden!e are there

from A to


5/13

!. Cf A%: # is a natural num-er, %: # is an even positive inteer andf: A is iven -y f(#) % 2#. Cs f a one$to$one !orresponden!e

In!erse of a Function

(#)%2#>1. 9(#)%2

1x. ind (2) and 9(). hat is your !on!lusion

3teps for ndin inverse fun!tion:1. rite y%f(#)2. Cnter!hane # with y vi!e versa.'. rite y as the fun!tion of #D this is the f$1(#).

"#$11: ind the inverse fun!tion of:

34.10

sin.9

5

32.8

ln.7

.6

log3.5

2.4

3)2(.3

2.2

32.1

2++=

=

+=

=

=

=

=

=

=

+=

xxy

xy

x

xy

xy

xey

xy

xy

xy

xy

xy

Inverse of a function might not be a function. Inverse of a functionis again a function if and only if the function is a onetoone

correspondence.

"#$12: hi!h one of the fun!tions in e#ample$11 has an inverse whi!h isalso a fun!tion

Com+osition of Functionsf:A, :E, h:E/f(#)%'#$*, (#)%2#, h(#)%@#Cs of % fo (!ommutative)Cs (ho)of (#) % ho(of) (#) (asso!iative)

ro+ert'# (fo)-% -of-.

Odd and E!en Functionsun!tion y % f(#) is odd if f($#) % $f(#). un!tion y % f(#) is even if f($#) % f(#).Fdd fun!tion: symmetri! a-out (B, B). "ven fun!tion: symmetri! a-out y a#is.

"#: hi!h one is odd and whi!h one is even


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a. f(#) % #2, -. f(#) % #', !. f(#) % sin #, d. f(#) % !os #, e. f(#) %tan #,

f. f(#) % #'.!os 2#.

iece.ise (iecemea/" Function

"#: /raw the pie!ewise (pie!emeal) fun!tion:

>

=1,2

1,)(

2

xx

xxxf


7/13

; % universal set A % fuy set A% mem-ership fun!tion B N AO 1, so A: ; PB, 1Q

Cf 1

1)(

2 += xx

A

nd (B), (1B), (1BB)Discrete Fu44' SetA %1Mx1>2Mx2>H..>n6xn

x1 ,x2 , H..xn : mem-ers of the setA1,2,H..n : deree of mem-ership"#:Lniversal set ;%&$,$*,$',$2,$1,B,1,2,',*, with deree of mem-ership areB, B.1, B.', B., B., 1, B., B., B.', B.1, B. rite the set notation of set

A%&real num-er !lose to ero I

Continuous Fu44' Set

Su++ort of a Fu44' Set 3upport A % supp A % set of mem-ers of universal set whose derees

are positive. "#: ;%&1,2,+,1',1,21,',B% set of people with various aes.

AesR

Eateories

a-y Ehild Teenaer Adults


8/13

"#: uy set of teenaers, nd teenaersB., teenaersB., teenaers1.ind younB.2, younB.?, youn1

Sca/ar Cardina/it': A%the sum of deree of mem-ership of ; in A."#: ind -a-y, !hild, teenaer, adult, youn, old

O+erations in Fu44' Set1. 3et in!lusion

A% if and only if de(A)%de()A if and only if de(A)de()"#:A % B.1M1 > B.'M2 > B.M' % B.2M1 > B.'M2 > B.M'Cs A % Cs A

2. Eomplement %AS%A!

AS%set of mem-ers with deree % 1$de(A)"#: ;%&1,2,+,1',1,21,',B. Fld%B.2M' > 1MB. ind the set of not

oldU.1M2>1M+>1M1'>1M1>1M21>B.2M'. ind the set of not

younU.Cs not old means youn Cs not youn means old

'. Lnion A: nd the ma#imum deree of A or ."#:A % B.1M1 > B.'M2 > B.*M' > B.2M* % B.2M1 > B.M2 > B.'M' > B.1Mind A

AesR

Eateories a-y Ehild Teenaer Adults


9/13

% B.2M1 > B.M2 > B.'M' > B.1Mind Aind youn old.

"#: ;%&1,2,',*,,+,

A%B.+M'>B.+M>B.2M+%B.M'>1M*>B.M+ind A, A, AS, S, ASS. ASS

Continuous Fu44' Set9ivin two fuy utilities e#pressed in the followins (# is real num-er)

>

B.M/. L % &A, , E, /, ", . /etermine

a. K!% KS-. X K!X% X KSX!. support of Kd. KB.

2. 4et fuy set A and iven -y (# is real num-er)

>2.

./raw the pie!emeal fun!tion:

+ B.+M > 1M+%B.2M1 > B.*M2 > B.?M' > 1M* > B.?M> B.*M+ > B.2Mind A, A, A!, !, A!!,A!!

E$ercise#1. U= {1, 2, 3, 4, , !, ", #, $, 1%&A= {2, 4, !, #, 1%&B= {1, 3, !, ", #&C= {3, "&

'a( I))*+-ae he +e+ U,A, Ba.d C/. a 0e.. d/ag-a, a-/.g a)) he e)ee.+ /. hea--/ae )ace+5 '6e7 /f a.y -eg/. /. y*- d/ag-a de+ . c.a/. a.y e)ee.+, -e8d-a9he +e )+ c--ec h/+5(

'b( Using your Venn diagram, )/+ he e)ee.+ /. each f he f))9/.g +e+7 B:A;, B: C;,A8 B

'c( C)ee he +aee. *+/.g a +/.g)e +yb)7 C8 B= 555 5

2.

3. A +*-?ey f 1%% +*de.+ -e?ea)ed1# )/e ea ch/ce.,4% )/e ea beef,2% )/e ea )ab,12 )/e ea bh ch/ce. a.d beef, )/e ea bh ch/ce. a.d )ab,4 )/e ea bh beef a.d )ab,3 )/e ea a)) h-ee5C)a++/fy a +*de. 9h de+ . )/e ea a.y f he h-ee /.d+ f ea a+ a ..8ea

eae-5'/( H9 a.y +*de.+ )/e ea a )ea+ .e f he /.d+ f ea>'//( H9 a.y ..8ea eae-+ a-e he-e>'///( H9 a.y +*de.+ )/e ea .)y )ab>

4. 9iven 29)( xxf = , xxg = 5)( ,12

2)(

+

=x

xxh .

a. nd domain and rane of ea!h.-. Cf m(#)%2#>', ind mo (#) and omof (#)


11/13

c5 F/.d h81'x(

Exercise

1. Lniversal set is real num-ers. /ene fuy sets A and withmem-ership fun!tions:

xxA += 11

)( and xxB 21

1

)( += .a. Cs A "#plain.-. Cs A % A "#plain.

x-3 -2 -1 1 2 3 4

y

-3

-2

-1

1

2

3

4

2. 9iven:

=

+=

0,3

})0{(,3)(

1

x

Rxxf

x

ind the inverse of f(#).

'. 9iven f(#) % *#'$ , # [, and (#) %x2

12 + , # [ $&B.

ind: fo(#) and of(#).*. \alidMinvalid

sq

rp

srqp

)()(

. \erify valid or invalid of the followin usin truth ta-le:

cb

a

bd

cd

da

+. There are * Cndonesians and * Cndians sittin around a round ta-le.a. 0ow many ways of sittin are there-. hat is the pro-a-ility that they sit a!!ordin to their nationality!. hat is the pro-a-ility that they sit alternately

. ind the domain and rane of:

a. 2xy= -. 3xy= !. xy = 16 d. 216 xy = e.1

42

+

=x

xy


12/13

?. ind the domain and rane of:

a. xy 2= -. xy log2= !.2

1

+

=x

xy d.

2

1

+

=x

xy

@. /raw a. 12 += xy -. xy 2= , for $2# 2.1B. ind the inverse of f(#) % #2, #B.

11. ind the inverse of f(#) % '(# $ 1)2> *, for # 1.12. ind the inverse of f(#) % 2#2>+#>12, for # $1..1'. /etermine odd or even or neither:

a. y%#.sin #-. y%#2.tan #!. y%#'.!os #d. y%2#

1*. /raw: 1=+ yx

EXERCISE

15 @e+ he ?a)/d/y f7a5

( )( )

rp

rpq

rqp

b5 If a/+ a. dd .*be- a.d b/+ a. e?e. .*be- he. 'a+ b) /+ a.+ dd .*be-5

25 F/.d he c.c)*+/. f he f))9/.g+5

a5 p q

p q

qq r

b5 q p

+ 3

p q

35 G/?e.7 f'9,x,y,( = 9xy 9xy 9xy 9xy 9xy 9xy 9xy 9xya5 D-a9 he )g/c gae

b5 E/)/fy

45 +e d/-ec -f7a5 @he +* f a. e?e. .*be- a.d a. dd .*be- /+ a. dd .*be-5b5 If 'a2( /+ a *)/)e f he. 'a28 4( /+ a *)/)e f 5

5 +e ahea/ca) /.d*c/. -?e7


13/13

a512)12()12(

1....

75

1

53

1

31

1

+=

+++

+

+

nn

nn, f- . 1

b5 n'> n /+ d/?/+/b)e by ! f- .1c5 42.$1 /+ d/?/+/b)e by 3 f- .1!5 -?e *+/.g ahea/ca) /.d*c/.7

a5)2(2

1)1(2

143

1)1(1....

141

131

121

2222 +

+=

+++

+

+

nnn , . /+ +//?e /.ege-5

b51

132

2

24

2

1.....

2

14

2

13

2

121

+=

++

+

+

+

n

nn

n , . /+ +//?e /.ege-5

"5a5 If C'x,2(= ! , f/.d xb5 H9 a.y /e cde+ ca. be b*/) f- 3 )ee- a.d 4 .*be-+ '-ee//. a-e

a))9edc5 H9 a.y I6 .*be-+ a-e he-e /f he I6 c.+/++ f a )ea+ 4 .*be- a.d a

+ ! .*be-+ 9/h* -ee//.5d5 @he-e a-e 4 e. a.d 4 9e.5 F*- f he a-e g/?e. +ch)a-+h/5 H9 a.y9ay+ a-e he-e /f a )ea+ 2 e. *+ ge he +ch)a-+h/+>

#5 F/.d he "he- a.d f/.d he e-+ c.a/./.g x"f

15

33 4

2

+

x

y

y

x5

$5 D-a9 he 0e..+ d/ag-a f-7a5 'A : B( 8 'A : B : C ( 9he-e A : B : C Jb5 'A : B ( 'B : C( 9he-e A : C = Jc5 'Ac: B ( C 9he-e B : C J

1%5 If K E K de.e+ he .*be- f ebe-+ f *./?e-+a) +e E, 9he-e E = { x K 1 L x L1%%, x /+ /.ege-&5 @he-e a-e 3 +*b+e+ f E, /5e5 A, B, C5 If K A K = '1% (,

K B K = '" 2(, K C K = '# 2(, K A : B K = '2 (, K A : C K = '3 (,

K B : C K = 2, K A : B : C K = , a.d K'A B C (K = 4%5F/.d a.d f/.d he .*be- f e)ee.+ f A, B, a.d C5

115 G/?e. E /+ he *./?e-+a) +e 9/h 3 +*b+e+, /5e5 A, B a.d C, 9he-e C A:B ,K A YB K = k !, K A 8 B K = 1, K B 8 A K = 1% k, K 'A:B( C K = ,K E 8 'A:B( K = 2%5 If K E K = 4k 1, f/.d k, KAK KBK, KCK da. KEK5

125 @he-e a-e 3 ?a-/e/e+ f /ce c-ea7 chc)ae, ?a./))a a.d +-a9be--y5 If +e.e

b*y+ h-ee +c+ f /ce c-ea, h9 a.y ch/ce+ de+ he ha?e>

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