INF1383 -Bancos de Dados

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "INF1383 -Bancos de Dados"

Transcription

1 3//0 INF383 -ncos de Ddos Prof. Sérgio Lifschitz DI PUC-Rio Eng. Computção, Sistems de Informção e Ciênci d Computção LGER RELCIONL lguns slides sedos ou modificdos dos originis em Elmsri nd Nvthe, Fundmentls of Dtse Systems, 4th Edition 004 Person Eduction, Inc. e Dtse System Concepts, McGrw Hill 5th Edition 005 Silerschtz, Korth nd Sudrshn INF383 D Sérgio Lifschitz Query Lnguges Lnguge in which user requests informtion from the dtse. Ctegories of lnguges Procedurl Non-procedurl, or declrtive Pure lnguges: Reltionl lger Tuple reltionl clculus Domin reltionl clculus Pure lnguges form underlying sis of query lnguges tht people use. INF383 D Sérgio Lifschitz Slide -56

2 3//0 Reltionl lger The sic set of opertions for the reltionl model is known s the reltionl lger. These opertions enle user to specify sic retrievl requests. The result of retrievl is new reltion, which my hve een formed from one or more reltions. The lger opertions thus produce new reltions, which cn e further mnipulted using opertions of the sme lger. sequence of reltionl lger opertions forms reltionl lger expression, whose result will lso e reltion tht represents the result of dtse query (or retrievl request). INF383 D Sérgio Lifschitz Slide 3-56 Reltionl lger Procedurl lnguge Five sic opertors select: σ project: union: set difference: Crtesin product: x INF383 D Sérgio Lifschitz Slide 4-56

3 3//0 Unry Reltionl Opertions SELECT Opertion SELECT opertion is used to select suset of the tuples from reltion tht stisfy selection condition. It is filter tht keeps only those tuples tht stisfy qulifying condition those stisfying the condition re selected while others re discrded. In generl, the select opertion is denoted y σ <selection condition> (R) where the symol σ (sigm) is used to denote the select opertor, nd the selection condition is oolen expression specified on the ttriutes of reltion R INF383 D Sérgio Lifschitz Slide 5-56 Select Opertion Nottion: σ p (r) p is clled the selection predicte Defined s: σ p (r) = {t t r nd p(t)} Where p is formul in propositionl clculus consisting of terms connected y : (nd), (or), (not) Ech term is one of: <ttriute> op <ttriute> or <constnt> where op is one of: =,, >,. <. INF383 D Sérgio Lifschitz Slide

4 3//0 Select Opertion Exmple C D Reltion r σ = ^ D > 5 (r) C D INF383 D Sérgio Lifschitz Slide 7-56 SELECT Opertion properties The SELECT opertion σ <selection condition> (R) produces reltion S tht hs the sme schem s R The SELECT opertion σ is commuttive; i.e., σ <condition> (σ < condition> ( R)) = σ <condition> (σ < condition> ( R)) cscded SELECT opertion my e pplied in ny order; i.e., σ <condition> (σ < condition> (σ <condition3> ( R)) = σ <condition> (σ < condition3> (σ < condition> ( R))) cscded SELECT opertion my e replced y single selection with conjunction of ll the conditions; i.e., σ <condition> (σ < condition> (σ <condition3> ( R)) = σ <condition> ND < condition> ND < condition3> ( R))) INF383 D Sérgio Lifschitz Slide

5 3//0 Unry Reltionl Opertions (cont.) PROJECT Opertion This opertion selects certin columns from the tle nd discrds the other columns. The PROJECT cretes verticl prtitioning one with the needed columns (ttriutes) contining results of the opertion nd other contining the discrded Columns. The generl form of the project opertion is π<ttriute list>(r) where π (pi) is the symol used to represent the project opertion nd <ttriute list> is the desired list of ttriutes from the ttriutes of reltion R. The project opertion removes ny duplicte tuples, so the result of the project opertion is set of tuples nd hence vlid reltion. INF383 D Sérgio Lifschitz Slide 9-56 Nottion: Project Opertion ( r ),, K, k where, re ttriute nmes nd r is reltion nme. The result is defined s the reltion of k columns otined y ersing the columns tht re not listed Duplicte rows removed from result, since reltions re sets INF383 D Sérgio Lifschitz Slide

6 3//0 Project Opertion Exmple Reltion r: C ,C (r) C C = INF383 D Sérgio Lifschitz Slide -56 PROJECT Opertion properties The numer of tuples in the result of projection π <list> (R) is lwys less or equl to the numer of tuples in R. If the list of ttriutes includes key of R, then the numer of tuples is equl to the numer of tuples in R. π <list> (π <list> (R) ) = π <list> (R) s long s <list> contins the ttriutes in <list> INF383 D Sérgio Lifschitz Slide -56 6

7 3//0 Reltionl lger Opertions From Set Theory UNION Opertion The result of this opertion, denoted y R S, is reltion tht includes ll tuples tht re either in R or in S or in oth R nd S. Duplicte tuples re eliminted. The two opernds must e type comptile. Type Comptiility The opernd reltions R (,,..., n ) nd R (,,..., n ) must hve the sme numer of ttriutes, nd the domins of corresponding ttriutes must e comptile; tht is, dom( i )=dom( i ) for i=,,..., n. INF383 D Sérgio Lifschitz Slide 3-56 Union Opertion Nottion: r s Defined s: r s = {t t r or t s} For r s to e vlid.. r, s must hve the sme rity (sme numer of ttriutes). The ttriute domins must e comptile (sme type of vlues) INF383 D Sérgio Lifschitz Slide

8 3//0 Union Opertion Exmple Reltions r, s: 3 r s r s: 3 INF383 D Sérgio Lifschitz Slide 5-56 Reltionl lger Opertions From Set Theory (cont.) Set Difference (or MINUS) Opertion The result of this opertion, denoted y R - S, is reltion tht includes ll tuples tht re in R ut not in S. The two opernds must e "type comptile. INF383 D Sérgio Lifschitz Slide

9 3//0 Set Difference Opertion Nottion r s Defined s: r s = {t t r nd t s} Set differences must e tken etween comptile reltions. INF383 D Sérgio Lifschitz Slide 7-56 Set Difference Opertion Exmple Reltions r, s: 3 r s r s: INF383 D Sérgio Lifschitz Slide

10 3//0 Reltionl lger Opertions From Set Theory (cont.) CRTESIN (or cross product) Opertion This opertion is used to comine tuples from two reltions in comintoril fshion. In generl, the result of R(,,..., n ) x S(,,..., m ) is reltion Q with degree n + m ttriutes Q(,,..., n,,,..., m ), in tht order. The resulting reltion Q hs one tuple for ech comintion of tuples one from R nd one from S. Hence, if R hs n R tuples (denoted s R = n R ), nd S hs n S tuples, then R x S will hve n R * n S tuples. The two opernds do NOT hve to e "type comptile INF383 D Sérgio Lifschitz Slide 9-56 Crtesin-Product Opertion Nottion r x s Defined s: r x s = {t q t r nd q s} ssume tht ttriutes of r(r) nd s(s) re disjoint. (Tht is, R S = ). If ttriutes of r(r) nd s(s) re not disjoint, then renming must e used. INF383 D Sérgio Lifschitz Slide

11 3//0 Crtesin-Product Opertion Exmple Reltions r, s: C D E r r x s: C D E s INF383 D Sérgio Lifschitz Slide -56 Renming ttriutes ttriute renming: chnge the nme of n ttriute δ (R) chnge to in reltion R Exmple: R(,) δ C (R) R (C,) is renmed to C in R, content is unchnged. INF383 D Sérgio Lifschitz Slide -56

12 3//0 Renming Reltions llows us to nme, nd therefore to refer to, the results of reltionl-lger expressions. llows us to refer to reltion y more thn one nme. Exmple: ρ x (E) returns the expression E under the nme X If reltionl-lger expression E hs rity n, then ρ x ( ( E ),,..., n ) returns the result of expression E under the nme X, nd with the ttriutes renmed to,,., n. INF383 D Sérgio Lifschitz Slide 3-56 ssignment Opertion The ssignment opertion ( ) provides convenient wy to express complex queries. Write query s sequentil progrm consisting of series of ssignments followed y n expression whose vlue is displyed s result of the query. ssignment must lwys e mde to temporry reltion vrile My use vrile in susequent expressions. INF383 D Sérgio Lifschitz Slide 4-56

13 3//0 Composition of Opertions nd ssignments Expressions using multiple opertions. Exmple: σ =C (r x s) () T r x s () Resp σ =C ( T ) C D E C D E INF383 D Sérgio Lifschitz Slide 5-56 R Expression: Forml Definition sic expression in reltionl lger (R) consists of either one of the following: reltion in the dtse constnt reltion Let E nd E e reltionl-lger expressions; the following re ll reltionl-lger expressions: E E ; E E ; E x E σ p (E ), P is predicte on ttriutes in E s (E ), S is list consisting of some of the ttriutes in E ρ x (E ), x is the new nme for the result of E INF383 D Sérgio Lifschitz Slide

14 3//0 dditionl Opertions dditionl opertions tht do not dd ny power to the reltionl lger, ut my simplify common queries. Set intersection Nturl join Division Generlized Projection ggregte Functions Outer Join INF383 D Sérgio Lifschitz Slide 7-56 Reltionl lger Opertions From Set Theory (cont.) INTERSECTION OPERTION The result of this opertion, denoted y R S, is reltion tht includes ll tuples tht re in oth R nd S. The two opernds must e "type comptile" INF383 D Sérgio Lifschitz Slide

15 3//0 Set-Intersection Opertion Nottion: r s Defined s: r s = { t t r nd t s } ssume: r, s hve the sme rity ttriutes of r nd s re comptile Note: r s = r (r s) INF383 D Sérgio Lifschitz Slide 9-56 Set-Intersection Opertion Exmple Reltion r, s: r r s 3 r s INF383 D Sérgio Lifschitz Slide

16 3//0 Reltionl lger Opertions From Set Theory (cont.) oth union nd intersection re commuttive opertions: R S = S R, nd R S = S R oth union nd intersection cn e treted s n-ry opertions pplicle to ny numer of reltions s oth re ssocitive: R (S T) = (R S) T, nd (R S) T = R (S T) The minus opertion is not commuttive; tht is, in generl R - S S R INF383 D Sérgio Lifschitz Slide 3-56 inry Reltionl Opertions JOIN Opertion The sequence of crtesin product followed y select is used quite commonly to identify nd select relted tuples from two reltions It is specil opertion, clled JOIN, denoted y This opertion is very importnt for ny reltionl dtse with more thn single reltion, ecuse it llows us to process reltionships mong reltions. The generl form of join opertion on two reltions R(,,..., n ) nd S(,,..., m ) is: R <join condition>s where R nd S cn e ny reltions tht result from generl reltionl lger expressions. INF383 D Sérgio Lifschitz Slide

17 3//0 inry Reltionl Opertions (cont.) EQUIJOIN Opertion The most common use of join involves join conditions with equlity comprisons only. Such join, where the only comprison opertor used is =, is clled n EQUIJOIN. In the result of n EQUIJOIN we lwys hve one or more pirs of ttriutes (whose nmes need not e identicl) tht hve identicl vlues in every tuple. NTURL JOIN Opertion ecuse one of ech pir of ttriutes with identicl vlues is superfluous, new opertion clled nturl join denoted y * ws creted to get rid of the second (superfluous) ttriute in n EQUIJOIN condition. The stndrd definition of nturl join requires tht the two join ttriutes, or ech pir of corresponding join ttriutes, hve the sme nme in oth reltions. If this is not the cse, renming opertion is pplied first. INF383 D Sérgio Lifschitz Slide Nturl-Join Opertion Let r nd s e reltions on schems R nd S respectively. Then, r s is reltion on schem R S otined s follows: Consider ech pir of tuples t r from r nd t s from s. If t r nd t s hve the sme vlue on ech of the ttriutes in R S, dd tuple t to the result, where t hs the sme vlue s t r on r t hs the sme vlue s t s on s Exmple: R = (,, 3, 4) e S = (,, 4) Result schem = (,, 3, 4, ) r s is defined s: r., r., r.3, r.4, s. (σ r. = s. r.4 = s.4 (r x s)) INF383 D Sérgio Lifschitz Slide

18 3//0 8 INF383 D Sérgio Lifschitz Slide Nturl Join Opertion Exmple Reltions r, s: δ 4 C D 3 3 D E δ r δ C D E δ s r s INF383 D Sérgio Lifschitz Slide Complete Set of Reltionl Opertions The set of opertions including select σ, project π, union, set difference -, nd crtesin product X is clled complete set ecuse ny other reltionl lger expression cn e expressed y comintion of these five opertions. For exmple: R <join condition> S = σ <join condition> (R X S)

19 3//0 inry Reltionl Opertions (cont.) DIVISION Opertion The division opertion is pplied to two reltions R(Z) S(X), where X suset Z. Let Y = Z - X (nd hence Z = X Y); tht is, let Y e the set of ttriutes of R tht re not ttriutes of S. The result of DIVISION is reltion T(Y) tht includes tuple t if tuples t R pper in R with t R [Y] = t, nd with t R [X] = t s for every tuple t s in S. For tuple t to pper in the result T of the DIVISION, the vlues in t must pper in R in comintion with every tuple in S. INF383 D Sérgio Lifschitz Slide Division Opertion Exmple Reltions r, s: δ δ δ r s: INF383 D Sérgio Lifschitz Slide

20 3//0 Division Opertion Suited to queries tht include the phrse for ll. r s Let r nd s e reltions on schems R nd S respectively where R = (,, m,,, n ) S = (,, n ) The result of r s is reltion on schem R S = (,, m ) r s = { t t R-S (r) u s ( tu r ) } Where tu mens the conctention of tuples t nd u to produce single tuple INF383 D Sérgio Lifschitz Slide nother Division Exmple Reltions r, s: C D E D E r 3 s r s: C INF383 D Sérgio Lifschitz Slide

21 3//0 Property Division Opertion (Cont.) Let q = r s Then q is the lrgest reltion stisfying q x s r Definition in terms of the sic lger opertion Let r(r) nd s(s) e reltions, nd let S R To see why r s = R-S (r ) R-S ( ( R-S (r ) x s ) R-S,S (r )) R-S,S (r) simply reorders ttriutes of r R-S ( R-S (r ) x s ) R-S,S (r) ) gives those tuples t in R-S (r ) such tht for some tuple u s, tu r. INF383 D Sérgio Lifschitz Slide 4-56 Generlized Projection Extends the projection opertion y llowing rithmetic functions to e used in the projection list. F E is ny reltionl-lger expression, F,..., F n ( E) Ech of F, F,, F n re re rithmetic expressions involving constnts nd ttriutes in the schem of E. INF383 D Sérgio Lifschitz Slide 4-56

22 3//0 dditionl Reltionl Opertions ggregte Functions nd Grouping type of request tht cnnot e expressed in the sic reltionl lger is to specify mthemticl ggregte functions on collections of vlues from the dtse. Exmples of such functions include retrieving the verge or totl slry of ll employees or the totl numer of employee tuples. These functions re used in simple sttisticl queries tht summrize informtion from the dtse tuples. Common functions pplied to collections of numeric vlues include SUM, VERGE, MXIMUM, nd MINIMUM. The COUNT function is used for counting tuples or vlues. INF383 D Sérgio Lifschitz Slide ggregte Functions nd Opertions ggregtion function tkes collection of vlues nd returns single vlue s result. vg: verge vlue min: minimum vlue mx: mximum vlue sum: sum of vlues count: numer of vlues ggregte opertion in reltionl lger ϑ ) ( E G, G, K, Gn F ( ), F (, K, Fn ( n ) E is ny reltionl-lger expression G, G, G n is list of ttriutes on which to group (cn e empty) Ech F i is n ggregte function Ech i is n ttriute nme INF383 D Sérgio Lifschitz Slide 44-56

23 3//0 ggregte Opertion nd Generlized Projection - exmples Reltion r: g sum(cc) (r) CC CP , (CC CP) (r) CC - CP 5 sum(cc ) INF383 D Sérgio Lifschitz Slide nking Dtse Exmple rnch (rnch_nme, rnch_city, ssets) customer (customer_nme, customer_street, customer_city) ccount (ccount_numer, rnch_nme, lnce) lon (lon_numer, rnch_nme, mount) depositor (customer_nme, ccount_numer) orrower (customer_nme, lon_numer) INF383 D Sérgio Lifschitz Slide

24 3//0 ggregte Opertion Exmple Reltion ccount grouped y rnch-nme: rnch_nme ccount_numer lnce Perryridge Perryridge righton righton Redwood rnch_nme g sum(lnce) (ccount) rnch_nme Perryridge righton Redwood sum(lnce) INF383 D Sérgio Lifschitz Slide ggregte Functions (Cont.) Result of ggregtion does not hve nme Cn use renme opertion to give it nme For convenience, we permit renming s prt of ggregte opertion rnch_nme g sum(lnce) s sum_lnce (ccount) INF383 D Sérgio Lifschitz Slide

25 3//0 Null Vlues It is possile for tuples to hve null vlue, denoted y null, for some of their ttriutes null signifies n unknown vlue or tht vlue does not exist. The result of ny rithmetic expression involving null is null. ggregte functions simply ignore null vlues (s in SQL) For duplicte elimintion nd grouping, null is treted like ny other vlue nd two nulls re ssumed to e the sme (s in SQL) INF383 D Sérgio Lifschitz Slide Null Vlues (cont.) Comprisons with null vlues return the specil truth vlue: unknown If flse ws used insted of unknown, then not ( < 5) would not e equivlent to >= 5 Three-vlued logic using the truth vlue unknown: OR: (unknown or true) = true, (unknown or flse) = unknown (unknown or unknown) = unknown ND: (true nd unknown) = unknown, (flse nd unknown) = flse, (unknown nd unknown) = unknown NOT: (not unknown) = unknown Result of select predicte is treted s flse if it evlutes to unknown INF383 D Sérgio Lifschitz Slide

26 3//0 dditionl Reltionl Opertions (cont.) The OUTER JOIN Opertion In NTURL JOIN tuples without mtching (or relted) tuple re eliminted from the join result. Tuples with null in the join ttriutes re lso eliminted. This mounts to loss of informtion. set of opertions, clled outer joins, cn e used when we wnt to keep ll the tuples in R, or ll those in S, or ll those in oth reltions in the result of the join, regrdless of whether or not they hve mtching tuples in the other reltion. INF383 D Sérgio Lifschitz Slide 5-56 dditionl Reltionl Opertions (cont.) Types of OUTER JOIN The left outer join opertion keeps every tuple in the first or left reltion R in R S; if no mtching tuple is found in S, then the ttriutes of S in the join result re filled or pdded with null vlues. similr opertion, right outer join, keeps every tuple in the second or right reltion S in the result of R S. third opertion, full outer join, denoted y keeps ll tuples in oth the left nd the right reltions when no mtching tuples re found, pdding them with null vlues s needed. INF383 D Sérgio Lifschitz Slide

27 3//0 Outer Join nking D Exmple (/4) Reltion lon lon_numer L-70 L-30 L-60 rnch_nme Downtown Redwood Perryridge mount Reltion orrower customer_nme lon_numer Jones Smith Hyes L-70 L-30 L-55 INF383 D Sérgio Lifschitz Slide Outer Join Exmple (/4) FIRST: recll the inner Join (nturl join) lon orrower lon_numer rnch_nme mount customer_nme L-70 L-30 Downtown Redwood Jones Smith INF383 D Sérgio Lifschitz Slide

28 3//0 Outer Join Exmple (3/4) Left Outer Join lon orrower lon_numer rnch_nme mount customer_nme L-70 L-30 L-60 Downtown Redwood Perryridge Jones Smith null Right Outer Join lon orrower lon_numer rnch_nme mount customer_nme L-70 L-30 L-55 Downtown Redwood null null Jones Smith Hyes INF383 D Sérgio Lifschitz Slide Outer Join Exmple (4/4) Full Outer Join lon orrower lon_numer rnch_nme mount customer_nme L-70 L-30 L-60 L-55 Downtown Redwood Perryridge null null Jones Smith null Hyes INF383 D Sérgio Lifschitz Slide

DATABASTEKNIK - 1DL116

DATABASTEKNIK - 1DL116 DATABASTEKNIK - DL6 Spring 004 An introductury course on dtse systems http://user.it.uu.se/~udl/dt-vt004/ Kjell Orsorn Uppsl Dtse Lortory Deprtment of Informtion Technology, Uppsl University, Uppsl, Sweden

More information

Learning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus

Learning Goals. Relational Query Languages. Formal Relational Query Languages. Formal Query Languages: Relational Algebra and Relational Calculus Forml Query Lnguges: Reltionl Alger nd Reltionl Clculus Chpter 4 Lerning Gols Given dtse ( set of tles ) you will e le to express dtse query in Reltionl Alger (RA), involving the sic opertors (selection,

More information

Reasoning and programming. Lecture 5: Invariants and Logic. Boolean expressions. Reasoning. Examples

Reasoning and programming. Lecture 5: Invariants and Logic. Boolean expressions. Reasoning. Examples Chir of Softwre Engineering Resoning nd progrmming Einführung in die Progrmmierung Introduction to Progrmming Prof. Dr. Bertrnd Meyer Octoer 2006 Ferury 2007 Lecture 5: Invrints nd Logic Logic is the sis

More information

Boolean Algebra. Boolean Algebra

Boolean Algebra. Boolean Algebra Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with: - two inry opertions, commonly denoted y + nd, - unry opertion, usully denoted y ˉ or ~ or, - two elements usully clled zero nd

More information

Handout: Natural deduction for first order logic

Handout: Natural deduction for first order logic MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes

More information

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018

Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Finite Automt Theory nd Forml Lnguges TMV027/DIT321 LP4 2018 Lecture 10 An Bove April 23rd 2018 Recp: Regulr Lnguges We cn convert between FA nd RE; Hence both FA nd RE ccept/generte regulr lnguges; More

More information

Harvard University Computer Science 121 Midterm October 23, 2012

Harvard University Computer Science 121 Midterm October 23, 2012 Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is

More information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. Comparing DFAs and NFAs (cont.) Finite Automata 2 CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

More information

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.) CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

More information

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb.

Types of Finite Automata. CMSC 330: Organization of Programming Languages. Comparing DFAs and NFAs. NFA for (a b)*abb. CMSC 330: Orgniztion of Progrmming Lnguges Finite Automt 2 Types of Finite Automt Deterministic Finite Automt () Exctly one sequence of steps for ech string All exmples so fr Nondeterministic Finite Automt

More information

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama

Homework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)

More information

CHAPTER 1 Regular Languages. Contents

CHAPTER 1 Regular Languages. Contents Finite Automt (FA or DFA) CHAPTE 1 egulr Lnguges Contents definitions, exmples, designing, regulr opertions Non-deterministic Finite Automt (NFA) definitions, euivlence of NFAs nd DFAs, closure under regulr

More information

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

More information

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

More information

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS

CS 310 (sec 20) - Winter Final Exam (solutions) SOLUTIONS CS 310 (sec 20) - Winter 2003 - Finl Exm (solutions) SOLUTIONS 1. (Logic) Use truth tles to prove the following logicl equivlences: () p q (p p) (q q) () p q (p q) (p q) () p q p q p p q q (q q) (p p)

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

Chapter 2 Finite Automata

Chapter 2 Finite Automata Chpter 2 Finite Automt 28 2.1 Introduction Finite utomt: first model of the notion of effective procedure. (They lso hve mny other pplictions). The concept of finite utomton cn e derived y exmining wht

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

set is not closed under matrix [ multiplication, ] and does not form a group.

set is not closed under matrix [ multiplication, ] and does not form a group. Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

More information

Fachgebiet Rechnersysteme1. 1. Boolean Algebra. 1. Boolean Algebra. Verification Technology. Content. 1.1 Boolean algebra basics (recap)

Fachgebiet Rechnersysteme1. 1. Boolean Algebra. 1. Boolean Algebra. Verification Technology. Content. 1.1 Boolean algebra basics (recap) . Boolen Alger Fchgeiet Rechnersysteme. Boolen Alger Veriiction Technology Content. Boolen lger sics (recp).2 Resoning out Boolen expressions . Boolen Alger 2 The prolem o logic veriiction: Show tht two

More information

Name Ima Sample ASU ID

Name Ima Sample ASU ID Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:35-9:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to

More information

INTRODUCTION TO LINEAR ALGEBRA

INTRODUCTION TO LINEAR ALGEBRA ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR

More information

CS 330 Formal Methods and Models

CS 330 Formal Methods and Models CS 0 Forml Methods nd Models Dn Richrds, George Mson University, Fll 2016 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 8 1. Prove q (q p) p q p () (4pts) with truth tle. p q p q p (q p) p q

More information

Riemann Integrals and the Fundamental Theorem of Calculus

Riemann Integrals and the Fundamental Theorem of Calculus Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums

More information

Introduction to Electrical & Electronic Engineering ENGG1203

Introduction to Electrical & Electronic Engineering ENGG1203 Introduction to Electricl & Electronic Engineering ENGG23 2 nd Semester, 27-8 Dr. Hden Kwok-H So Deprtment of Electricl nd Electronic Engineering Astrction DIGITAL LOGIC 2 Digitl Astrction n Astrct ll

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

Context-Free Grammars and Languages

Context-Free Grammars and Languages Context-Free Grmmrs nd Lnguges (Bsed on Hopcroft, Motwni nd Ullmn (2007) & Cohen (1997)) Introduction Consider n exmple sentence: A smll ct ets the fish English grmmr hs rules for constructing sentences;

More information

Lecture 2 : Propositions DRAFT

Lecture 2 : Propositions DRAFT CS/Mth 240: Introduction to Discrete Mthemtics 1/20/2010 Lecture 2 : Propositions Instructor: Dieter vn Melkeeek Scrie: Dlior Zelený DRAFT Lst time we nlyzed vrious mze solving lgorithms in order to illustrte

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

Lecture 9: LTL and Büchi Automata

Lecture 9: LTL and Büchi Automata Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled

More information

Section 6.1 Definite Integral

Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

expression simply by forming an OR of the ANDs of all input variables for which the output is

expression simply by forming an OR of the ANDs of all input variables for which the output is 2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

Math 017. Materials With Exercises

Math 017. Materials With Exercises Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson

More information

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q. 1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTORS. 4.0 Introduction. Objectives. Activity 1 4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

More information

3 Regular expressions

3 Regular expressions 3 Regulr expressions Given n lphet Σ lnguge is set of words L Σ. So fr we were le to descrie lnguges either y using set theory (i.e. enumertion or comprehension) or y n utomton. In this section we shll

More information

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations

Resources. Introduction: Binding. Resource Types. Resource Sharing. The type of a resource denotes its ability to perform different operations Introduction: Binding Prt of 4-lecture introduction Scheduling Resource inding Are nd performnce estimtion Control unit synthesis This lecture covers Resources nd resource types Resource shring nd inding

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

Chapter 2. Random Variables and Probability Distributions

Chapter 2. Random Variables and Probability Distributions Rndom Vriles nd Proilit Distriutions- 6 Chpter. Rndom Vriles nd Proilit Distriutions.. Introduction In the previous chpter, we introduced common topics of proilit. In this chpter, we trnslte those concepts

More information

Math 554 Integration

Math 554 Integration Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we

More information

Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrals. Partitioning the Curve. Estimating the Mass Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

More information

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018

CS 301. Lecture 04 Regular Expressions. Stephen Checkoway. January 29, 2018 CS 301 Lecture 04 Regulr Expressions Stephen Checkowy Jnury 29, 2018 1 / 35 Review from lst time NFA N = (Q, Σ, δ, q 0, F ) where δ Q Σ P (Q) mps stte nd n lphet symol (or ) to set of sttes We run n NFA

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.

Calculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved. Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

More information

Anatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute

Anatomy of a Deterministic Finite Automaton. Deterministic Finite Automata. A machine so simple that you can understand it in less than one minute Victor Admchik Dnny Sletor Gret Theoreticl Ides In Computer Science CS 5-25 Spring 2 Lecture 2 Mr 3, 2 Crnegie Mellon University Deterministic Finite Automt Finite Automt A mchine so simple tht you cn

More information

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010 CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

More information

Homework Solution - Set 5 Due: Friday 10/03/08

Homework Solution - Set 5 Due: Friday 10/03/08 CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.

More information

Software Engineering using Formal Methods

Software Engineering using Formal Methods Softwre Engineering using Forml Methods Propositionl nd (Liner) Temporl Logic Wolfgng Ahrendt 13th Septemer 2016 SEFM: Liner Temporl Logic /GU 160913 1 / 60 Recpitultion: FormlistionFormlistion: Syntx,

More information

STRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada

STRUCTURE OF CONCURRENCY Ryszard Janicki. Department of Computing and Software McMaster University Hamilton, ON, L8S 4K1 Canada STRUCTURE OF CONCURRENCY Ryszrd Jnicki Deprtment of Computing nd Softwre McMster University Hmilton, ON, L8S 4K1 Cnd jnicki@mcmster.c 1 Introduction Wht is concurrency? How it cn e modelled? Wht re the

More information

Formal Languages and Automata Theory. D. Goswami and K. V. Krishna

Formal Languages and Automata Theory. D. Goswami and K. V. Krishna Forml Lnguges nd Automt Theory D. Goswmi nd K. V. Krishn Novemer 5, 2010 Contents 1 Mthemticl Preliminries 3 2 Forml Lnguges 4 2.1 Strings............................... 5 2.2 Lnguges.............................

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

Intuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras

Intuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore-632014, TN, Indi

More information

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch.

Overview HC9. Parsing: Top-Down & LL(1) Context-Free Grammars (1) Introduction. CFGs (3) Context-Free Grammars (2) Vertalerbouw HC 9: Ch. Overview H9 Vertlerouw H 9: Prsing: op-down & LL(1) do 3 mei 2001 56 heo Ruys h. 8 - Prsing 8.1 ontext-free Grmmrs 8.2 op-down Prsing 8.3 LL(1) Grmmrs See lso [ho, Sethi & Ullmn 1986] for more thorough

More information

MA Handout 2: Notation and Background Concepts from Analysis

MA Handout 2: Notation and Background Concepts from Analysis MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

More information

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15

Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15 Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

MTH 505: Number Theory Spring 2017

MTH 505: Number Theory Spring 2017 MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

CS 330 Formal Methods and Models

CS 330 Formal Methods and Models CS 330 Forml Methods nd Models Dn Richrds, section 003, George Mson University, Fll 2017 Quiz Solutions Quiz 1, Propositionl Logic Dte: Septemer 7 1. Prove (p q) (p q), () (5pts) using truth tles. p q

More information

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.

More information

Continuous Random Variables

Continuous Random Variables STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

More information

Sturm-Liouville Theory

Sturm-Liouville Theory LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

1.3 Regular Expressions

1.3 Regular Expressions 56 1.3 Regulr xpressions These hve n importnt role in describing ptterns in serching for strings in mny pplictions (e.g. wk, grep, Perl,...) All regulr expressions of lphbet re 1.Ønd re regulr expressions,

More information

Arithmetic & Algebra. NCTM National Conference, 2017

Arithmetic & Algebra. NCTM National Conference, 2017 NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

More information

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-*

Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Regular Expressions (RE) Kleene-* Regulr Expressions (RE) Regulr Expressions (RE) Empty set F A RE denotes the empty set Opertion Nottion Lnguge UNIX Empty string A RE denotes the set {} Alterntion R +r L(r ) L(r ) r r Symol Alterntion

More information

CS375: Logic and Theory of Computing

CS375: Logic and Theory of Computing CS375: Logic nd Theory of Computing Fuhu (Frnk) Cheng Deprtment of Computer Science University of Kentucky 1 Tle of Contents: Week 1: Preliminries (set lger, reltions, functions) (red Chpters 1-4) Weeks

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

ECON 331 Lecture Notes: Ch 4 and Ch 5

ECON 331 Lecture Notes: Ch 4 and Ch 5 Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve

More information

1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automata 1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

More information

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information

WENJUN LIU AND QUÔ C ANH NGÔ

WENJUN LIU AND QUÔ C ANH NGÔ AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous

More information

z TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability

z TRANSFORMS z Transform Basics z Transform Basics Transfer Functions Back to the Time Domain Transfer Function and Stability TRASFORS Trnsform Bsics Trnsfer Functions Bck to the Time Domin Trnsfer Function nd Stility DSP-G 6. Trnsform Bsics The definition of the trnsform for digitl signl is: -n X x[ n is complex vrile The trnsform

More information

Lexical Analysis Finite Automate

Lexical Analysis Finite Automate Lexicl Anlysis Finite Automte CMPSC 470 Lecture 04 Topics: Deterministic Finite Automt (DFA) Nondeterministic Finite Automt (NFA) Regulr Expression NFA DFA A. Finite Automt (FA) FA re grph, like trnsition

More information

ENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions

ENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner non-repeted ctors. The cover-up rule my be used: 4 4 4

More information

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints)

Chapter 4 Regular Grammar and Regular Sets. (Solutions / Hints) C K Ngpl Forml Lnguges nd utomt Theory Chpter 4 Regulr Grmmr nd Regulr ets (olutions / Hints) ol. (),,,,,,,,,,,,,,,,,,,,,,,,,, (),, (c) c c, c c, c, c, c c, c, c, c, c, c, c, c c,c, c, c, c, c, c, c, c,

More information

8 Laplace s Method and Local Limit Theorems

8 Laplace s Method and Local Limit Theorems 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved

More information

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS 1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

More information

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon

Energy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,

More information

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.) MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

More information

dy ky, dt where proportionality constant k may be positive or negative

dy ky, dt where proportionality constant k may be positive or negative Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

More information

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY

Chapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in

More information

1 From NFA to regular expression

1 From NFA to regular expression Note 1: How to convert DFA/NFA to regulr expression Version: 1.0 S/EE 374, Fll 2017 Septemer 11, 2017 In this note, we show tht ny DFA cn e converted into regulr expression. Our construction would work

More information

Formal languages, automata, and theory of computation

Formal languages, automata, and theory of computation Mälrdlen University TEN1 DVA337 2015 School of Innovtion, Design nd Engineering Forml lnguges, utomt, nd theory of computtion Thursdy, Novemer 5, 14:10-18:30 Techer: Dniel Hedin, phone 021-107052 The exm

More information

The Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2

The Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2 The Bernoulli Numbers John C. Bez, December 23, 2003 The numbers re defined by the eqution e 1 n 0 k. They re clled the Bernoulli numbers becuse they were first studied by Johnn Fulhber in book published

More information

9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes

9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen

More information

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t

More information

Designing Information Devices and Systems I Spring 2018 Homework 8

Designing Information Devices and Systems I Spring 2018 Homework 8 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices:

Signal Flow Graphs. Consider a complex 3-port microwave network, constructed of 5 simpler microwave devices: 3/3/009 ignl Flow Grphs / ignl Flow Grphs Consider comple 3-port microwve network, constructed of 5 simpler microwve devices: 3 4 5 where n is the scttering mtri of ech device, nd is the overll scttering

More information