CS233601: Discret e CS233601: Discret e CS233601: Discret e Mat - - PDF document

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CS233601: Discret e Mat hemat ics CS233601: Discret e CS233601: Discret e Mat hemat ics Mat hemat ics

Depart ment of Comput er Science Depart ment of Comput er Science Nat ional Nat ional Tsing Tsing Hua Hua Universit y Universit y

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‧I nst ruct or

–Shun-Ren Yang (楊舜仁), sr yang@cs.nt hu.edu.t w –Of f ice Number : R3202, EECS Building

‧Time and Locat ion

–Wednesday 10:10 ~ 12:00, Fr iday 11:10 ~ 12:00 –EECS 236

‧Text book

–"Discr et e Mat hemat ics and I t s Applicat ions" (McGr aw- Hill), by Kennet h H. Rosen

‧Ref erence

–"Element s of Discret e Mat hemat ics" (McGr aw-Hill), by Pr of . C. L. Liu

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‧Requirement s

–Homework (30%) –Examinat ions x 4 (60%) –Part icipat ion (10%)

‧Teaching Assist ant s

–顏勝盈, 綜二館741, 分機3542, siyan@wmnet .cs.nt hu.edu.t w –黃鼎鈞, 綜二館741, 分機3542, superd1230@gmail.com

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What is Discret e Mat hemat ics?

‧Discret e mat hemat ics is t he part of mat hemat ics devot ed t o t he st udy of discret e obj ect s. ‧Here discret e means consist ing of dist inct

• r unconnect ed element s.

‧The kind of problems solved using discret e mat hemat ics include:

–How many ways are t her e t o choose a valid passwor d on a comput er syst em?

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What is Discret e Mat hemat ics? (Cont .)

–What is t he pr obabilit y of winning a lot t ery? –I s t her e a link bet ween t wo comput er s in a net work? –What is t he shor t est pat h bet ween t wo cit ies using a t r anspor t at ion syst em? –How can a list of int egers be sort ed so t hat t he int egers ar e in incr easing order ? –How many st eps ar e requir ed t o do such a sort ing? –How can it be pr oved t hat a sor t ing algor it hm corr ect ly sort s a list ?

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What is Discret e Mat hemat ics? (Cont .)

–How can a cir cuit t hat adds t wo int egers be designed? –How many valid I nt ernet addresses ar e t here?

‧You will learn t he discret e st ruct ures and t echniques needed t o solve such problems. ‧More generally, discret e mat hemat ics is used whenever obj ect s are count ed, when relat ionships bet ween f init e (or count able) set s are st udied, and when processes involving a f init e number of st eps are analyzed.

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Why St udy Discr et e Mat hemat ics?

‧Through t his course you can develop your mat hemat ical mat urit y, t hat is, your abilit y t o underst and and creat e mat hemat ical argument s. ‧Discret e mat hemat ics provides t he mat hemat ical f oundat ions f or many comput er science courses, including dat a st ruct ures, algorit hms, dat abase t heory, aut omat a t heory, f ormal languages, compiler t heory, comput er securit y, and operat ing syst ems.

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Five Themes in Discr et e Mat hemat ics?

‧Mat hemat ical Reasoning

–Under st anding mat hemat ical r easoning in or der t o read, compr ehend, and const r uct mat hemat ical ar gument s –Mat hemat ical logic, met hods of pr oof , mat hemat ical induct ion

‧Combinat or ial Analysis

–The abilit y t o count or enumer at e obj ect s –The basic t echniques of count ing, per mut at ions, combinat ions

‧Discr et e St r uct ur es

–The abst r act mat hemat ical st r uct ur es used t o r epr esent discr et e obj ect s and r elat ionships bet ween t hese obj ect s –Set s, per mut at ions, r elat ions, gr aphs, t r ees, and f init e-st at e machines

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Five Themes in Discr et e Mat hemat ics?

‧Algorit hmic Thinking

–The specif icat ion of an algor it hm t hat solves cer t ain classes of problems –The specif icat ion of t he algor it hm, t he ver if icat ion

• f t he algor it hm, t he analysis of t he space and t ime

complexit ies of t he algor it hm

‧Applicat ions and Modeling

–Comput er science, dat a net working, chemist r y, business, et c. –Modeling wit h discr et e mat hemat ics is an ext remely impor t ant problem-solving skill

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Course Out line

‧The Foundat ions: Logic and Proof , Set s, and Funct ions ‧The Fundament als: Algor it hms, t he I nt egers, and Mat rices ‧Mat hemat ical Reasoning, I nduct ion, and Recursion ‧Count ing ‧Discret e Probabilit y

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Course Out line (Cont .)

‧Advanced Count ing Techniques ‧Relat ions ‧Graphs ‧Trees ‧Boolean Algebra ‧Modeling Comput at ion