MELJUN CORTES MANUAL Discrete Structure CSCI08

8/13/2019 MELJUN CORTES MANUAL Discrete Structure CSCI08

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CSCI08 Instructional Manual

Prepared by: ELJUN P. CORTES 2nd trimester, SY 2007-2008

Course Title: DISCRETE STRUCTURES I

Course Description:

The course introduces the students to mathematical logic: propositions qualifiers,predicates, proof, techniques; mathematical induction; Fundamentals of set theory: sets,power sets, algebra of sets, relations, functions, accountability and finiteness; graphsand trees.

Course Objectives:

At the end of the term, the students should be ble to:

a. To apply the theories and operations on sets.b. To understand and appreciate the use of combinations and

progression for practical application.c. To understand and apply relations, digraphs, and functions by way of

problems applicable to computer applications.

!RE"I# !ERIOD

SETS_ It is any well-defined collection of objects.

_It is a collection of distinct objects, without repetition and without ordering

Set $ottion:enoted by capital letter.

!"ample:

# $ %, , &' ( $ %), *, +'

$ %a, b, c, d' $ %ascal, (, (obol'

#ethods In Describin% Set:

). Listing Method /oster 0 Tabular Form

_It is listing or enumerating all the elements of a gi1en set.!"ample: # $ %a, b, c'

*. Rule Method 2et-uilder Form

_The elements ha1e properties in common._The elements must satisfy a gi1en rule or condition.

!"ample: # $ % " 3 " is a positi1e integer 4 5'# $ % " 3 " 4 6'

College of Computer Studies Prepared by: ELJUN P. CORTES

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Element_It is a member of a gi1en set or an object in the collection

Element $ottion:enoted by small letter0s.

!"ample:

# $ %a, b, c' where: a # ; b # ; c # ; d #

T&pes of Set

). Finite elements can be counted.!"ample: # $ %a, b, c'

*. Infinite elements cannot be counted.!"ample: # $ % " 3 " is a positi1e integer'

Thin%s to remember in listin% elements:

). 7rder of elements is not important*. /epetition must be ignored

Subset ' (_If # is a subset of , # is contained in or e1ery element of # is in .

!"ample:# $ %a, b' $ %a, b, c'

# but #$ote:

). !1ery set is a subset of itself*. #n empty set is a subset of e1ery set.

3. If # then there is at least one element in # that is not in .

!"ample: # $ %), *, + '

The subsets of # are:8#9 $ % %)', %*', %+', %), *', %), +', %*, +', %), *, +', % ' '

To chec:

3#3 $ *n$ *+$


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!roper Subset '(_# is a proper sunset of if # and #

Crdinlit& of the set ' ) ) (

_It is the total number of elements in a gi1en set

!"ample: # $ %), *, + '3 # 3 $ +

Empt& Set * $ull or +oid Set

_It is a set containing no element

!"ample: # $ % ' or # $

If # $ %', # is not an empty set

Sin%leton

_a set containing only one element!"ample: # $ % ) '

Comprble Set

_#t least one set is a subset of another set

(onditions so that # < are comparable:

1. # < #

!"ample: # $ %), *, +, 5' $ %5, +, *, )'

2. # < #

!"ample: #$ %), * ' $ %), *, +'

3. # < #

#$ %a, b, c' $ %a, b'


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$onComprble Set

_=o set is a subset of the other set. 8 # < # < # 9

!"ample: # $ %), *, +'

$ %5, 6, >'

E-ul Set_It is a set ha1ing e"actly the same cardinality and ind of elements.

!"ample: # $ %), *, +' $ %), *, +'

E-uivlent Set

_It is a set ha1ing the same cardinality.

!"ample: # $ %), *, +' $ %5, 6, >'

Onetoone correspondence

_!1ery element of # is associated or paired to e1ery element in .

!"ample: # $ %), *, +' $ %5, 6, >'

Disjoint Set

_The sets ha1ing no common elements

!"ample: # $ %), 5, 6' $ %+, *, >'

Universl Set or Set of ll Sets

_It is the biggest set under in1estigation wherein the other sets are subsets of thegi1en set.


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!"ample: ? $ %), *, +, 5, 6, >' @ =7 /!!TITI7=# $ %), *, +' $ %5, 6, >'( $ %6, >'

Set of Sets, .mil& of Sets, Clss of Sets

_It is a collection of set

_#ll the elements of the gi1en set are sets.

!"ample: # $ %#, , ('# $ %%), *, +', %5, 6, >', %6, >'' $ %#, , (, , &, *'( $ %#, , (, , A'

A Complement /ith Respect to: 0, A1, AC

U new set of elements that belong to ? but not in # 8 with respect to ? 9!"ample: ? $ %), *, +, 5, 6, >'

# $ %), *, +'#B $ %5, 6, >'

2 new sets of elements that belong to but not in # 8with respect to 9!"ample: $ %a, b, c, d, e'

# $ %a, b, c'#B $ %d, e'

!o/er Set '! 'A((

_It is a family of all subsets of a gi1en set

_It is the set of all subsets

!"ample: # $ %), *, +'8#9 $ % %)', %*', %+', %),*', %*, +', %),+', % ', %), *, +' '

To chec: 8#9 $ *n

$ *+

$

2ASIC SET O!ERATIO$S

34 Union 'A 2( _It is a new set of elements that belong to # or or to both # and .!"ample: # $ %), *, +, 5'

$ %*, 5, >, C'

# $ %), *, +, 5, 6, >, C'


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b. # # $ #

;4( !roperties of $ull Seta. # $

b. # $ #

=4( !roperties of Universl Seta. # ? $ ?

b. # ? $ #

>4( !roperties of A Complementa. # $ # in1olution

b. # #B $ ?c. # #B $

d. B $ ?

e. ?B $ E E

f. # $ #

g. # $ #

!rovin%:). et:

? $ %), *, +, 5, 6, >'# $ %), +, 6'

$ %*, 5, >'( $ %5, 6'

De #or%nss "/ EEEEE E E

1. # $ #

%), *, +, 5, 6, >' $ %*, 5, >' %), +, 6'

%), *, +, 5, 6, >' $ %), *, +, 5, 6, >' EEEEE E E

2. # $ #

% ' $ %*, 5, >' %), +, 6'

% ' $ % '

*. et:? $ %a, b, c, d, e, f'

# $ %a, b, c' $ %c, d, e, f'

1 # #B $ ?


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# #



%a, b, c' %d, e, f' $ %a, b, c, d, e, f'

%a, b, c, d, e, f' $ %a, b, c, d, e, f'

+enn Di%rm

_It represents relation and operator using the plane geometrical figures such asrectangle, circle, ellipse.

# #

# #

# #


#

8

#


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Set Reltions

1. ( 2. #

3. #

4. (5. # (6. (

7. ( #

8. #

D. # and are not compatible)&. # and are not disjoint)). # and ( are disjoint

)*. and ( are not disjoint13.#, , ( ?

"I$E DIA


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34 T/o Sets Additiona9 isjoint

) A 2 ) ? ) A ) @ ) 2 )b9 =ot disjoint

) A

2 ) ? ) A ) @ ) 2 ) ) A 2 )

$ote:3 # 3 is subtracted from 3 # 3 G 3 3 to a1oid duplication because# belongs to both sets and the sum 3 # 3 G 3 3 includes the numbers of

elements twice.

!"ample:

# computer company must hire *6 programmers to handle systems programmingtass and 5& programmers for application programming. 7f those hired, )& will bee"pected to perform tas of each type. How many programmers must be hired

Ji1en: 3 # 3 $ *63 3 $ 5&

3 # 3 $ )&

2olution:

3 # 3$ 3 # 3 G 3 3 - 3 # 3

$ *6 G 5& )&$ 66 programmers

54 Three Sets Addition) A U 2 U C ) ? ) A ) @ ) 2 ) @ ) C ) ) A 2 ) ) A C) ) A C ) @ )A 2 C )

!"ample K):

# sur1ey is taen on methods of commuter tra1el. !ach respondent is to checbus, train or automobile as a major method of tra1eling to wor. Lore than one answer ispermitted. The result were as follows:

3 # 3 $ +& people checed bus

3 3 $ +6 people checed train3 ( 3 $ )&& people checed automobile

3 # 3 $ )6 checed bus and train

3 # ( 3 $ )6 checed bus and automobile

3 ( 3 $ *& checed train and automobile

3 # ( 3 $ 6 checed all three methods


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#6

6

)&)6

( C&



uestions:a.9 How many respondents completed the sur1eyb.9 How many checed e"actly one of the three methods of commuter tra1elc.9 raw the Menn diagram.

Answer:

a.9 3 # ? ? ( 3 $ 3 # 3 G 3 3 G 3 ( 3 - 3 # 3 - 3 ( 3 - 3 # (3 G 3 # ( 3

$ +& G +6 G )&& )6 *& )6 G6 $ )*&

b.9 3 # 3 $ )6 6 $ )& 3 # 3 $ +& )& )& 6 6 $ 6

3 ( 3 $ *& 6 $ )6 3 3 $ +6 )& )6 6 $ 6

3 # ( 3 $ )6 6 $ )& 3 ( 3 $ )&& )6 )& 6 $ C&

------------- & respondents

c.9 Menn iagram

!"ample K*:

2uppose that )&& (2 students tae at least one of the languages #2I(,(77, #2(#.

3 # 3 $ >6 study #2I( 3 # 3 $ *& study #2I( and (77

3 3 $ 5* study (77 3 # ( 3 $ *6 study #2I( and #2(#

3 ( 3 $ 5* study #2(# 3 ( 3 $ )6 study (77 and #2(#

uestions:

a. Find the number of students who study all of the languages.b. Find the number of students who study e"actly one of the languages.c. raw the Menn diagram representation.

Answer:

a. 3 # ? ? ( 3 $ 3 # 3 G 3 3 G 3 ( 3 - 3 # 3 - 3 ( 3 - 3 # ( 3 G 3 # ( 3


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)&

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#*

)

)* C

( )&



)&& $ >6G56G5*-*&-)6G" )&& $ )6& >& $ " " $ )6& >& )6*

" $

b. 3 # 3 $ *& $ )*

3 # ( 3 $ *6 $ )5

3 ( 3 $ )6 $ C

3 # 3 $ >6 )* )C $ * 3 3 $ 56 )* C $ ) 3 ( 3 $ 5* )C C $ )&

*&

c. Menn iagram

CO#2I$ATORICS!roduct Set

#. In1ol1ing * 2ets 8# N 9

_The product set of # and consist of ordered pair 8a, b9 where at a # and

b , denoted by # N 8# cross 9

_It is also called the (artesian roduct of # and

_# N $ %8a, b9 3 a #, b '

$ote:). # N $ %a, b' elements

2. # N N # not commutati1e unless # $ or one of the sets isempty

+. # N # $ #*

!"ample:# $ %), *, +' $ %a, b'


12

)*

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# N $ %8), a9, 8), b9, 8*, a9, 8*, b9, 8+, a9, 8+, b9' N # $ %8a, )9, 8a, *9, 8a, +9, 8b, )9, 8b, *9, 8b, +9'

# N # $ %8), )9, 8), *9, 8), +9, 8*, )9, 8*, *9, 8*, +9, 8+, )9, 8+, *9, 8+, +9'

24 Involvin% 6 Sets 'A B 2 B C(Ohere a #, b , c (

!"ample:# $ %a, b' $ %), *, +'( $ %", y'

# N N ( $ %8a, ), "9, 8a, ), y9, 8a, *, "9, 8a, *, y9, 8a, +, "9, 8a, +, y9, 8b, ), "9, 8b, ), y9, 8b, *, "9, 8b, *, y9, 8b, +, "9, 8b, +, y9'

Tree Di%rm

) " 8a, ), "9y 8a, ), y9

a * " 8a, *, "9y 8a, *, y9

+ " 8a, +, "9y 8a, +, y9

) " 8b, ), "9y 8b, ), y9

b * " 8b, *, "9y 8b, *, y9

+ " 8b, +, "9y 8b, +, y9

E+E$T _it is any happening or occurrence.

SUCCESSI+E E+E$TS _it is an occurrence, which can happen one after the other

SI#U"TA$EOUS E+E$TS_these are e1ents, which can happen at the same time

.U$DA#E$TA" !RI$CI!"ES O. COU$TI$ ways

c4 Combintion of Sum Rule nd !roduct Rule

!"ample:

How many different license plates are there that includes ), *, or + lettersfollowed by 5 digits

a. Oith repetition!) $ *>.)&.)&.)&.)& *>&,&&&!* $ *>.*>.)&.)&.)&.)& >,C>D,&&&!+ $ *>.*>.*>.)&.)&.)&.)& )C6,C>&,&&&


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)*,C&,&&& waysb. without repetition !) $ *>.)&.D..C )+),&5& !* $ *>.*6.)&.D..C +,*C>,&&& !+ $ *>.*6.*5.)&.D..C C,>*5,&&&

*,&+),&5& ways

.ACTORIA" $OTATIO$

.ctoril_It is the product of all integers from ) to nn ' n fctoril (_It is the product of all positi1e integers from ) to n inclusi1e when n isa positi1e integer.

#ethods:). for/rd 6P $ ).*.+.5.6*. bc/rd 6P $ 6.5.+.*.)

!ER#UTATIO$ _it is an arrangement of objects in a gi1en order#(, #(, #(, (((#, (#, (#

CASES of !ER#UTATIO$

I4 "iner !ermuttion nn, 8n,n9, nr, 8n,r9

A. Number if permutations of n distinct objects taen all a time

.ormul: n!n ? n

!"amples:

). In how many ways can teaching assistants be assigned to sectionsn $ $ P

$ .C.>.6.5.+.*.)$ 5&,+*& ways

*. Find the number of permutations of the word F/7L taen at all times.n $ 5


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55 $ 5P$ 5.+.*.)*5 ways

!. Number of permutation of n distinct objects taen r at a time

.ormul: n!r ? n

'n r(

!"amples:

). Find the number of permutation of > distinct objects taen + at a time.n $ > r $ +

>+ $ >P >.6.5.+P -------- $ -------------- $ )*& ways 8> - +9P$ +P

*. Find the number of permutation of the word F/7L taen * at a time.n $ 5 r $ *

5* $ 5P 5.+.*P ------ $ ------------ )* ways

85 - *9P *P

" . Number of #ermutations with repetition.

_It is the number of distinct permutation of n distinct objects where n) is one ind,

n* is another ind, n+is the third ind, nis the th ind.

.ormul: n! 'n1, n2, n3, nk( ? n n1 $2 $3 $k

!"ample:

Find the number of permutation of the word #Q.n) $ + 68+,),)9 $ 6P 6.5.+P

n*# $ ) -------------- $ ---------------- $ *&

n+Q $ ) +P*P)P +P

n $ 6


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II4 Circulr !ermuttion _It is an ordered arrangement of things in a circular manner orin a ring.

.ormul:n3

!n3

? ' n3 (

!"ample:

In how many ways can 5 people be seated at a round table

5-)5-) $ 85-)9P

$ +P$ >

CO#2I$ATIO$ _It is choosing or selecting of objects without regard of order.

n(r, (8n,r9, 8 n 9, (rn

A. Number of combinations taen r at a time.ormul: nCr ? n

r 'nr(

!"amples:

). In how many ways can a student answer D out of )& questions in an e"aminationn$)& r$D

n(r $ )&P )&PDP------------------ $ -------------- $ )& ways DP8)&-D9P DP)P

*. In how many committees can * chemists and + biologists be formed from 6 chemistsand > biologists

$ 6(* $ >(+6P >P

$ ---------------- " ----------------- *P8+9P +P8*P9

$ 6.5.+P >.6.5.+P -------------------------- " --------------------------


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*P +P +P *P$ )& " *&$ *&& ways

!. "ombination taen in a series

_The total number of permutations of n things taen at a time

.ormul: C3 @ nC5 @ nC6 @ nCn

!"ample:

In how many ways can one or more than 6 people enter a room

$ 5n7 3 $ 6() G 6(* G 6(+ G 6(5 G 6(6

$ *6 ) or $ 6 G )& G )& G 6 G )$ +) ways $ +) ways

#R$LIM $%AMINA&I'N

#IDTER# !ERIOD

#ATFE#ATICA" !RO


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_It is a sequence of the numbers called terms, each of which after the first deri1ed fromthe preceding one by adding to it a fi"ed number called common difference8d9, whichcan be found by subtracting any term from the one following it.

$lements of Arithmetic #rogressiona or a

)- first term

l or an - last termd - common differencen - number of termss - sum of all terms

Formulas:l ? 3@ 'n3(d

n nS ? '3 @ l ( or S ? H53 @ 'n3(d

5 5

!"amples:

). Orite the )st> terms of # ifa9 a $ )6 d $ -*

)6, )+, )), D, C, 6

b9 a $ -*& d $ 5

-*&, -)>, -)*, -, -5, &

*. Find the 1alue of

a9 )6, )), $ C b9 *, *+, $ ) c9 )C, , *+ $ *&

Arithmetic #en 'A#( _terms between the first and the last term of a #.


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!"ample:

Find the #L between a $ *, d $ *, l $ )&.

*, 5, >, , )&

Sum of ll terms

!"ample:

Find the sum of # if ), +, 6, R ais gi1en.

2 $ n0* S*a G 8n-)9d $ 0* S*8)9 G 8-)9* $ 58* G )59 $ >5

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*, 5, , )>, +*, >5

*. Find the * JL between 5 and +*. Find 2.5, , )>, +*

l $ arn-) 2 $ a rl

+* $ 5r + ) rr+ $ +* 0 5 $ 5 *8+*9

r+ $ ) *r $ * 2 $ >&

C4 Frmonic !ro%ression 'F!(_The sequence of numbers whose reciprocal forms an #.

!"ample: )06, )0C, )0D

Frmonic #en 'F#( _It is the reciprocals of the #L of the reciprocals of the 1aluesunder obser1ation.!"ample:

Insert * HL between )06 and )0)56, , )), )5

l $ a G 8n-)9d)5 $ 6 G 85-)9d)5 $ 6 G +d D $ +d d $ +

)06, 3*J, 3*33, )0)5

RE"ATIO$S A$D DI


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+. 8",y9 or 8a,b9 - open sentence where 8a,b9 is true or false for any orderedpair 8a,b9 belong to # N .

- It defines the relation of # to .

Reltion on A_It is a relation of # into itself, that is, a subset of # N #.

!"ample: # $ %), *, +' /elation on # $ %8), )9, 8), *9, 8), +9'

Open sentence 7 !'K,&( or !',b(

). !',b( is true a / b_a / b is related to b

!"ample:

et / $ %/,B/B, 8",y9' where / is a set of real numbers and 8",y9 reas U" is less thanyV

86,)&9 $ T 6 / )&

*. !', b( is flse # / b_a is not related to b

!"ample:

et / $ %/, W/B, 8",y9' where /B is a set of real numbers and 8",y9 reads U" isless than yV, that is Ua is less than bV

8)&,69 $ F )& / 6

2inr& Reltion from A to b_# relation that is a subject of # N , the first coordinates is from #, the secondcoordinates is from .

Reltion from A to 2 is subset of A B 24


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!"ample: / $ %8), *9, 8*, +9, 8*, 59'

/-)$ %8*, )9, 8+, *9, 85, *9'

!rtition or uotient Set_It is a collection of non-empty set subsets of # such that:

). !ach element of # belongs to one of the sets in .

2. If #)and #*are distinct elements of then #) #*$

locs or cells of partitions the sets in $ %#), #*, #+, #5, #6, #>, #C'

!"ample: et # $ %a, b, c, d, e, f, g, h'#) $ %a, b, c, d'#* $ %a, c, e, f, g, h'#+ $ %a, c, e, g'#5 $ %b, d'#6 $ %f, h'

). %#), #*' $ not *. %#), #6' $ not +. %#+, #5, #6' $ 5. %#*, #5' $

Set Arisin% from Reltion R

34 Domin of R or dom R_2et of the first coordinate of /.

54 Rn%e of R or rn R

_2et of the second coordinate of /.

!"ample: et # $ %), *, +' $ %+, 5, 6'/ $ %8),*9, 8*, +9, 8), +9'

dom / $ %), *'ran /) $ %*,+'


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Reltive set of K R'K(_It is the set of all y in with the properties that is " is related to y.

!"ample: et : # $ %), b, c, d' / $ %8a, a9, 8a, b9, 8b, c9, 8c, a9, 8a, c9, 8c, b9'

Find:). /8a9 $ %a, b, c'*. /8b9 $ %c'+. /8c9 $ %a, b'5. /8d9 $ % '

R Reltive Set of A3 R'A3(

_It is the set of all y in b with the property that is /-related to y for some " in # )

$ote: /8#)9 is the union of the sets /8"9 where " #.

!"ample: etermine /8#)9 and /8#*9 if

#)$ %a, b' and #*$ %c, d'

Find:). /8#)9 $ %a, b, c'

*. /8#*9 $ %a, b'

MI)&$RM $%AMINA&I'N

.I$A" !ERIOD

#triK of Reltion #R

_It is a rectangular array of numbers arranged in m horiAontal rows and n 1erticalcolumns.

$ote: & if 8a, b9 /) if 8a, b9 /

!"amples:


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). et #$ %), *, +' $ %r, s'/ $ %8), r9, 8*, s9, 8+, r9'

LR$ ) && )) &

*. et # $ %a), a*, a+' $ %b), b*, b+, b5'/ $ %8a), b)9, 8a), b59, 8a*, b*9, 8a*, b+9, 8a+, b)9, 8a+, b+9'

LR$) & & )& ) ) &) & ) &

Di%rphs or Directed


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)

*

+

5

)

*

+

5

)

*

+

5



!"ample K*: # $ %), *, +, 5'/ $ %8), )9, 8), *9, 8*, )9, 8*, *9, 8+, +9, 85, 59, 8+, 59, 85, +9, 8*, +9, 8), 59, 8*, 59'

In 7 De%ree_It is the number of edges terminating the 1erte".Out 7 De%ree_It is the number of edges lea1ing the 1erte".

Restriction of R to 2 ? R '2 K 2(

!"ample:# $ %a, b, c, d, e, f' $ %a, b, c'/ $ %8a, c9, 8b, c9, 8c, d9, 8a, e9, 8c, e9'" $ %8a, a9, 8a, b9, 8a, c9, 8b, a9, 8b, b9, 8b, c9, 8c, a9, 8c, b9, 8c, c9'

/estriction of / to $ %8a, c9, 8b, c9'


27

+ertices ) * + 5

In * * + 5

7ut + 5 * *


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) * +

6 5



!th Reltions nd Di%rphs

!th "en%th_In / from a to b is finite sequence beginning with a and ending with b.

!"ample:

) path length is 5, 1erte" ) to +

) $ ), *, 6, 5, + $ ), *, *, 5, + $ ), *, *, *, +

* path length is + 1erte" ) to )* $ ), *, 6, )

+ path length is ) 1erte" ) to *

+ $ ), *

!th Composition8/efer to e"ample of path length9

* o ) $ ) G * $ ), *, 6, 5, + G ), *, 6, )

$ no path composition

) o * $ * G ) $ ), *, 6, ) G ), *, 6, 5, +

$ ), *, 6, ), *, 6, 5, +

+ o * $ * G + $ ), *, 6, ) G ), * $ ), *, 6, ), *

* o + $ + G * $ ), * G ), *, 6

$ no path composition

C&cle_a path that begins and ends at the same 1erte"!"ample: *

n 7 r& reltion 'Rn(

_If n is a fi"ed positi1e integer, we define a relation /nas:


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) * +

5 6 >

) * +

5 6 >



" /ny means that there is a path length n from " to y in /.

/)$ path length of ) connecting the 1ertices

/*$ path length of * connecting the 1ertices

/+$ path length of + connecting the 1ertices

/n$ path length of 5 connecting the 1ertices

!"ample: et # $ %), *, +, 5, 6, >'/ $ %8), *9, 8), 59, 8*, *9, 8*, 59, 8*, 69, 8+, 59, 85, 69, 86, >9'

igraph of /

/*$ %8), *9, 8), 59, 8*, *9, 8*, 59, 8*, 69, 8+, 59, 85, 69, 86, >9'

igraph of /*

Find the / 8connecti1ity relation for /9


2!

a b

c d

+


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/ $ %8a, a9, 8a, b9, 8a, c9, 8a, d9, 8a, e9, 8b, c9, 8b, d9, 8b, e9, 8c, d9'

!roperties of Reltion

). RefleKive_e1ery element of # is related to itself.!"ample: # $ %), *, +'

/ %8), )9, 8*, *9, 8+, +9'

IrrefleKive_no element of # is related to itself.!"ample: # $ %), *, +'

/ $ %8), *9, 8*, +9, 8+, )9, 8), +9, 8+, *9, 8*, )9'

*. S&mmetric_whene1er 8a,b9 is in /, 8b,a9 is also in /.!"ample: # $ %), *, +'

/ $ %8), *9, 8*, +9, 8+, )9, 8), +9, 8+, *9, 8*, )9'

Ass&metric_whene1er 8a,b9 is in /, 8b, a9 is not in /.!"ample: # $ %), *, +'

/ $ %8), +9, 8*, +9, 8), *9'

+. Trnsitive_whene1er 8a,b9 is in /, 8b,c9 is also in /, 8a,c9 is in / also.!"ample: # $ %), *, +'

/ $ %8), *9, 8*, +9, 8), +9, 8), )9, 8*, *9, 8+, +9'

$ote: whene1er / is refle"i1e, the gi1en / is also transiti1e.

E-uivlnce Reltions_It is a relation in set # is an equi1alence / if

a. relation is refle"i1eb. relation is symmetricc. relation is transiti1e


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!"ample: # $ %), *, +'/ $ %8), *9, 8*, +9, 8+, *9, 8*, )9, 8), +9, 8), )9, 8+, )9, 8*, *9, 8+, +9'

E-uivlence of R determined b& !

!"ample K): et # $ %), *, +, 5' $ %%), *, +', %5''

!qui1alence of / determined by $ %8), )9, 8), *9, 8), +9, 8*, )9,8*, *9, 8*, +9, 8+, )9, 8+, *9, 8+, +9, 85, 59'

!"ample K*: et # $ %), *, +, 5' $ %%), *', %+, 5''

!qui1alence of / determined by $ %8), )9, 8), *9, 8*, )9,8*, *9, 8+, +9, 8+, 59, 85, +9, 85, 59'

E-uivlence clssesA*R_It is a partition of set # determine by the equi1alence relation / on #. 2ets within aset.

!"ample: et # $ %), *, +, 5'/ $ %8), )9, 8), *9, 8*, )9, 8*, *9, 8+, +9, 8+, 59, 85, +9, 85, 59'

Find #0/

/8)9 $ %), *' /8+9 $ %+, 5'/8*9 $ %), *' /859 $ %+, 5'

#0/ $ %8), *9, 8+, 59'

#A$I!U"ATIO$ O. RE"ATIO$ EEComplementr& R or R

a / b if a / b


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/ 2 $ a8/ 29b $ a/b and a2b

/ 2 $ a8/ 29b $ a/b and a2b

Inverse R R3

_It is a relation from to # 8a re1erse order from /9b /-)a if a / b

!"ample K):

et # $ %), *, +, 5' $ %a, b, c'/ $ %8), a9, 8), b9, 8*, b9, 8*, c9, 8+, b9, 85, a9'2 $ %8), b9, 8*, c9, 8+, b9, 85, b9'

#" $ %8), a9, 8), b9, 8), c9, 8*, a9, 8*, b9, 8*, c9, 8+, a9, 8+, b9, 8+, c9, 85, a9, 85, b9, 85, c9'

Find: EE)./ $ %8), c9, 8*, a9, 8+, a9, 8+, c9, 85, b9, 85, c9'

*. / 2 $ %8), b9, 8*, c9, 8+, b9'

+./ 2 $ %8), a9, 8), b9, 8*, b9, 8*, c9, 8+, b9, 85, a9, 85, b9'

5. /-)$ %8a, )9, 8b, )9, 8b, *9, 8c, *9, 8b, +9, 8a, 59'6. 2 $ %8), a9, 8), c9, 8*, a9, 8*, b9, 8+, a9, 8+, c9, 85, a9, 85, c9'

>. 2-)$ %8b, )9, 8c, *9, 8b, +9, 8b, 59'

!"ample K*:

et # $ %a, b, c, d, e'

/ 2

Find:E). / $ %8a, a9, 8a, c9, 8b, a9, 8b, b9, 8c, a9, 8c, b9, 8c, d9, 8c, e9, 8d, a9, 8d, b9, 8d, e9, 8e, a98e, b9, 8e, c9, 8e, d9'


32

b

a c

e d

b

a c

e d


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*. / 2 $ %8a, b9, 8b, e9, 8c, c9'

2. / 2 $ %8a, a9, 8a, b9, 8a, d9, 8a, e9, 8b, b9, 8b, c9, 8b, d9, 8b, e9, 8c, b9, 8c, c9,

8c, e9,8d, b9, 8d, c9, 8d, d9, 8e, a9, 8e, d9, 8e, e9'

5. /-)$ %8b, a9, 8d, a9, 8e, a9, 8c, b9, 8d, b9, 8e, b9, 8c, c9, 8c, d9, 8d, d9, 8e, e9'6. 2 $ %8a, c9, 8a, d9, 8a, e9, 8b, a9, 8b, c9, 8b, d9, 8c, a9, 8c, d9, 8d, a9, 8d, c9, 8d, d9,

8d, e9,8e, b9, 8e, c9, 8e, e9'>. 2-)$ %8a, a9, 8b, a9, 8b, b9, 8e, b9, 8b, e9, 8c, c9, 8e, c9, 8b, d9, 8a, e9, 8d, e9'

!"ample K+:et # $ %), *, +'

L/$ ) & ) L2$ & ) )

& ) ) ) ) && & & & ) &

Find:

). L/$ & ) & 5. L/2$ ) ) )

) & & ) ) )) ) ) & ) &

*. L/2$ & ) ) 6. L2-) & ) &

) ) & ) ) )& ) & ) & &

+. L/-)$ ) & & >. L2$ ) & &

& ) & & & )) ) & ) & )

Composition

Composition of R nd S 'S o R(_It is a relation from # to (

_It is a is in # and c is in ( then a82o/9c if for some b in , we ha1e a/b and b/c.

$ote: L2 o / $ L/o L2

!"ample K):


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et # $ %), *, +,5'/ $ %8), *9, 8), )9, 8), +9, 8*, 59, 8+, *9'2 $ %8), 59, 8), +9, 8*, +9, 8+, )9, 85, )9'

2 o / $ %8), +9, 8), 59, 8), )9, 8*, )9, 8+, +9'/ o 2 $ %8), *9, 8*, *9, 8+, *9, 8+, )9, 8+, +9, 85, *9, 85, )9, 85, +9'

!"ample K*:

L/$ ) & ) L2$ ) & &

) ) ) & ) )& ) & ) & )

Find:

). L2o/$ ) & )

) ) )& ) )

/)-() /*-(* /+-() /)-(* /*-(* /+-(*) ) ) ) & ) ) & ) & & )& & ) & ) & & ) ) ) ) )) ) ) ) & ) ) & ) & & &

/) (+ /* (* /+ ()) & ) & & && ) ) ) ) )) ) ) ) & )

*. L/o2$ ) & )) ) )) ) )

/) () /* () /+ () /) (* /* (* /+ (*) ) ) ) ) ) ) & & & ) && ) & ) & ) & ) ) ) ) && & ) & ) & & ) ) ) ) )/) (+ /* (+ /+ (+) ) & ) ) )& ) ) ) & )) & ) & ) &


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.U$CTIO$S

.unctions or #ppin% or Trnsformtion

.unctions_To each element in #, there is assigned a unique element of .

_It is a special class or relation.

Note:=ot all relations can be considered as a function.

_# binary relation / from # to is said to be a function if for e1ery element a in #, thereis a unique element b in so that 8a,b9 is in /. For function f from # to , instead of

writing 8a,b9/, the notation f8a9 $ b is also used where b is the image of a.

_It is a #, then the element belonging to set which is assigned to a is called the

image of a and is denoted by f8a9, which is said as Uf of aV

_f denotes the assignment of / from # to

f: # $ f is function from # to $ f maps # into

f: #

Conditions for .unctions

). For e1ery element a #, only one element b is assigned.

*. #ll elements in # must be used up by the functions.+. =ot all elements in must be used up by the functions.5. The function may pair more than one input a to the same output b.

!"ample:). a & *. a &

b ) b )c * c * f not a function

+. a & 5. a &

b ) b )c * c *

not a function not a function

6. a & >. a &b ) b )c * c * f f


35

(o-domain or range offunction f

relation

omain of function f

function f


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M& of Definin% .unctionet # $ %), *, +, 5' $ %a, b, c, d'

34 ReltionF $ %8),a9, 8*,c9, 8+,b9, 85,d9'

54 Correspondencef8)9 $ a f8+9 $ bf8*9 $ c f859 $ d

+. iagram). .a ) a ) a

*. .b * b * b+. .c + c + c

5. .d 5 d 5 d

# # #

94 Rule

# ) a* b

+ c5 d

;4 .ormulf8"9 $ " G )

Identif& .unction or Identit& Trnsformtion on A

_For a function f : # # defined by the formula f8"9 $ ", let f assign to eachelement in # the element itself.


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f f f


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Opertor or Trnsformtion on A_a function of # into itself

_if the domain and co-domain of a function f are both the same set

f : # #

!"ample: et # $ %), *, +' ) ) * * + +

f # #

E-ul .unction_If f and g are functions defined on the same domain d and f8a9 $ g8a9 for e1ery a d,then the functions f and g are equal.

!"ample:

et function f and g be defined by the diagram.) )

) * ) + * + * D D

f g

f8)9 $ ) g8)9 $ )f8*9 $ D g8*9 $ D

f $ g


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Rn%e of .unction f ? f'(

_It consists precisely of those elements in which appear as the image of at least oneelement in set #._It consists of all the image points.=ote: f8#9

!"ample.1. et f: #

# $ %a,b,c,d' $ %a,b,c'efine the function f of # into by the correspondence f8a9 $ b,

f8b9 $ c, f8d9 $ b, then f8#9 $ %b,c'

*. et # $ %), *, +, 5, 6'efine the function f # # by the diagram

) ) f8#9 $ %*, +, 6'* *+ +5 56 6 f

# #

+. et # $ %-*, -), &, ), *'et the function g: # /K be defined by the formula g8"9 $ "*G )Find the range of a function g.

g8-*9 $ f g8)9 $ *

g8-)9 $ * g8*9 $ 6 g8#9 $ %), *, 6'g8&9 $ )

Constnt .unction_It is the same element b belonging to set is assigned to e1ery element in set #.

f8#9 $ one element from

!"ample:

et the function be defined by the diagrama ) f8#9 $ *

b *c +

f#


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Even .unction_If # is a set of nos. such that whene1er a #, -a # also and if f: # is a function

satisfying f8a9 $ f8-a9 for each a #.

!"ample:

) a -) a

* b -* b

Inverse of function f3'b(_It consists of those elements in # which ha1e b as their image._f)8b9 % " 3 " #, f8"9 $ b'_f)8b9 #

!"ample:a " f)8"9 $ %b'b y f)8y9 $ 8y,A9c A f)8A9 $ % ' f

#

EKtended Definition_It consists of those elements in # which are mapped into some elements in the domain.

!"ample:" r f)8r,s9 $ %",y'y s f)8r,t9 $ %y, A'A t f)89 $ f)8r,s,t9 $ %",y,A' f

#

Inverse function of f f 32 A_It must be both one-to-one and onto.

!"ample:a " " a a "b y y b or b y

c " A c c A f f) f)

# . # #


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Specil T&pes of .unction

). Onetoone '3:3( injection no two elements in # ha1e the same image.

!"ample:) 5 ) 5* 6 * 6+ > + > f f

# # not one-to-one function

*. Onto function e1ery element of is the image of one or more elements of #.

$ote:f8#9 $

!"ample:a ) a "b * b yc + c A f d

# f#

not one-to-onebut onto function

+. Onetoone nd ontofunction bijection!"ample:

a )b *c + f

#

Composition .unction of !roduct .unction

f: # and g: (

f g

8g o f9

_a function # into (.


40

# (

omain ofg

(o-domain ofg(o-domain off

domain off


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) )&* ))+ )* h o g o f

#

*. f o g o hC )& 5 C ) 5 )) 6 * 6D )* > D + > h g

( ( # =o product function

+. 8f o g o h9-)

no in1erse product function

5. 8h o g o f9-)

) )&* ))+ )*8h o g o f9-)

#

6. 8h o g o f98#9

$ %)&,)),)*'

). Formulaet f assign to each real number its square. For e1ery real number ", let f8"9 $ "*.The domain and co-domain are both real number.

f: /K / KThe image of -+ $ D

f8-+9 $ D or f: -+ D

*. Jeographicalet f assign to each country in the world its capital. omain of f $ set of countries

in the world. co-domain of f $ list of capital cities in the world.The image of France $ aris

f8France9 $ aris

+. (orresponceet # $ %a,b,c,d' $ %a,b,c'

f8a9 $ b f8b9 $ c f8c9 $ d


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5. iagramet # $ %a,b,c,d' $ %",y,A'

a "

b y

c Ad

#

6. /elation

et # $ %), *, +, 5' $ %a, b, c, d'f $ %8), a9, 8*, a9, 8+, d9, 85, c9'f8)9 $ a f8*9 $ a f8+9 $ d f859 $ c

>. et # $ %), *, +' $ %", y, A'/ $ %8), "9, 8*, "9'2 $ %8), "9, 8), y9, 8*, A9, 8+, y99

/ 2) " ) "* y * y+ " + A

FINAL $%AMINA&I'N

*******************************************************************************************************************

AMA COMPUTER UNIVERSITYProject 8, Quezon Cit

CO!!E"E O# COMPUTER STU$IES

MANUA!IN


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$ISCRETE STRUCTURE

MELJUN CORTES MANUAL Discrete Structure CSCI08

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