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CS233601: Discret e CS233601: Discret e CS233601: Discret e Mat hemat ics Mat hemat ics Mat hemat ics Depart ment of Comput er Science Depart ment of Comput er Science Nat ional Tsing Tsing Hua Hua Universit y Universit y Nat ional

  1. CS233601: Discret e CS233601: Discret e CS233601: Discret e Mat hemat ics Mat hemat ics Mat hemat ics Depart ment of Comput er Science Depart ment of Comput er Science Nat ional Tsing Tsing Hua Hua Universit y Universit y Nat ional 1 ‧ 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 2

  2. ‧ 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 3 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 or 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? 4

  3. 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 ? 5 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. 6

  4. 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. 7 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 8

  5. 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 of 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 9 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 10

  6. Course Out line (Cont .) ‧ Advanced Count ing Techniques ‧ Relat ions ‧ Graphs ‧ Trees ‧ Boolean Algebra ‧ Modeling Comput at ion 11

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