Discrete Structures (CSCI 2824, Spring 2015)

Course Information
  • Class Timings: MWF 10:00 - 10:50 AM

  • Class Location: ECCR 245

  • Instructor: Yuen-Lam Voronin

  • Pre-Requisites: Data Structures (CSCI 2270)

  • Office Hours:

    • Agile office hours: after class on MWF. Instructor will provide quick help to students who stay behind after class.

    • Regular office hours: Monday 3-4 PM, Thursday 10 AM-12 PM and 4.45-5.45 PM at ECOT 527.

      *NEW* additional hours: Wednesdays 5-6 PM

  • Course textbook: The course material will be drawn from Discrete Mathematics: Mathematical Reasoning and Proof with Puzzles, Patterns and Games by Douglas E. Ensley and J. Winston Crawley (2006).

    • The purchase of this textbook is optional, though encouraged if you are unfamiliar with the material in this course and/or need a good collection of exercise problems for practice.

    • Lecture notes will be posted online before each lecture, and the relevant book section number will be provided.

    • There is a 2-hour reserve copy in Gemmill; call no.: QA 9.25 .E57 2006.

    • Beware! The “paperback” version of the book is the student solutions manual. Please make sure that you are not purchasing the student solutions manual. We will not need it for the course.

  • Final Exam: Sunday May 3, 7:30 PM - 10:00 PM (Source: CU Boulder Academic Final Exam Schedule)

News
  • [Final review] Final review material will be available very shortly. Here are some concepts that you should know and practice questions.

  • [Office hours] The office hour from 4.45 to 5.45 on Thursday April 30 is cancelled. The morning session is as usual though.

  • [Written assignment] No more written assignment will be given. (So Assignment 9 is the last written assignment of this course.)

  • [Moodle] Moodle assignment 12 posted, due on Fri May 1.

Assignments

There are two components: weekly written assignments (posted below) and weekly Moodle assignments (posted on Moodle).

  • Each written assignment consists of 3 questions and occasionally 1 bonus question.

    Each Moodle assignment consists of 10-15 multiple choice questions.

  • Written assignments are due on Fridays @ 10 AM in class (starting from Jan 23).

    Moodle assignments are due on Tuesdays @ 4 PM Wednesdays @ 11.59 PM .

No late submissions will be accepted for any reason other than documented medical emergencies.

We will omit two assignments with the lowest scores while computing the overall grade.

Assignment Date posted Due date Topics covered Files
1 Jan 17 Jan 23 Sequences (Sect. 1.2), summation/product notations (Sect. 1.2) Problem set
Solution
2 Jan 31 Feb 6 Propositional logic (Sect. 1.3) and predicate logic (Sect. 1.4-1.5) Problem set
Solution
3 Feb 7 Feb 13 Direct proof techniques (Sect. 2.1-2.2) and Weak induction (Sect. 2.3) Problem set
Worksheet style
Some tips
Solution
4 Feb 15 Feb 20 Weak induction (Sect. 2.3) Strong induction (Sect. 2.4) Problem set
Worksheet style)
Solution
5 Mar 1 Mar 6 Basic notions in set theory (Sect. 3.1-3.3) Problem set
Worksheet style
Solution
6 Mar 8 Mar 13 Functions: injectiveness and surjectiveness (Sect. 4.1-4.3) Problem set
Worksheet style
Some examples
Solution
7 Mar 25 Apr 3 Growth of functions (Sect. 4.8) Problem set
Worksheet style
Solution
8 Apr 4 Apr 10 Basic counting techniques (Sect 5.1-5.3) Problem set
Worksheet style
Solution
9 Apr 18 Apr 2410 C(n,k), recursive counting, expectation (Sect 5.3, 5.6, 6.1) Problem set
Worksheet style
Examples Solution

Quizzes and Final Exam

There will be 4 20-minute quizzes throughout the semester, held on Fridays in class. (And on those Fridays there will not be any written assignment due.) The topics covered in each quiz will be available two weeks ahead.

No alternate exams will be offered unless there is a very compelling and documented personal or medical emergency. If you need special accommodations of any nature, you are expected to inform the instructor well in advance.

The lowest quiz will be omitted when computing the overall grade.

Quiz Date Topics covered Files
1 Friday Jan 30 Sequences (Sect. 1.2), summation/product notations (Sect. 1.2),
propositional logic and truth table (Sect. 1.3), predicate logic (Sect. 1.4)
Practice questions and solution
Quiz version 1 and its solution
Quiz version 2 and its solution
2 Friday Feb 27 Quantifiers (Sect. 1.4), implications (Sect. 1.5),
direct proof techniques including proof by cases and proving the contrapositive statements (Sect 2.1-2.2; see also online notes)
weak induction (Sect. 2.3),
strong induction (Sect. 2.4),
proof by contradiction and pigeonhole principle (Sect 2.5), set definition and operations (Sect. 3.1)
Practice questions and solution
Quiz Quiz and its solution
3 Friday Mar 20 Lecture materials from Feb 25, and between Mar 2 and Mar 13. Specifically:
Relations and functions (Sect 4.1),
injective/surjective/bijective functions, inverses, cardinality (Sect 4.2-4.3),
equivalence relations etc (Sect 4.4)
Practice questions and solution
Quiz version 1 and its solution
Quiz version 2 and its solution
4 Friday Apr 17 Lecture materials from Mar 16 to Apr 10. Specifically:
Growth of functions and the master theorem(Sect 4.8)
Basic combinatorial structures (Sect 5.1)
Basic rules for counting (Sect 5.1-5.2)
Permutation and combination (Sect 5.3)
Counting ordered/unordered lists (Sect 5.4)
Recursive counting (Sect 5.5)
Solving recursions (Sect 5.6)
Basic probability, up to the sum rule and complement rule (Sect 6.1-6.2)
Practice questions and solution
Quiz version 1 and its solution
Quiz version 2 and its solution
Final Sunday May 3, 7.30-10pm Everything in the course concepts that you should know
practice questions
solutions to nonproof questions
solutions to proof questions and T4(b)

Again, the final examination date is: Sunday May 3, 7:30 PM - 10:00 PM. One needs at least 25% grade from the final exam to pass the course.

Lecture Schedule and Notes

This course covers the following topics:

  • Logic: Propositional and Predicate Logic, Quantifiers.

  • Proofs: Primer on writing proofs, inductions, proof by contradiction.

  • Sets, Relations and Functions: Basic properties. Infinite sets. Cardinality.

  • Recursion: Recursive functions and recursively defined structures.

  • Combinatorics: Counting, binomial theorem, counting with recursion.

  • Probability: probability of discrete events, expectation.

  • Trees and graphs: Definitions and properties.

  • Interesting Applications: Cryptography (rsa) and Networks (social network analysis).

Pre-Lecture Work

Before you attend each lecture, we hope that you will spend 10-15 minutes to prepare, by reading the relevant portion of the book stated on the lecture schedule (at least skimming through the book or notes provided).

Some lectures may contain a small 5-10 minute tutorial problem that we will solve as a class. After solving the problem, you are expected to submit it to the instructor. The resulting grade will count for your course participation.

Lecture Schedule

The schedule of lectures shown below is subject to change. We will post notes for most topics before each lecture. Please take a look through them, and come prepared for class.

ID Date Topics Covered Book Sections Lecture Notes (Click on Links)
1 Jan 12 Binary Numbers 2.6 Binary Numbers
2 Jan 14 Sequences 1.2 Sequences
3 Jan 16 Summation 1.2 Summation
Examples on sequences and summation
Jan 19 No class: Martin Luther King day
4 Jan 21 Propositional Logic: Propositions, Connectives and Truth Tables 1.3 Propositional logic
5 Jan 23 Predicate Logic: Predicates and quantifiers 1.4 Predicate logic
6 Jan 26 Predicate Logic: quantifiers and implication 1.4-1.5 Quantifiers and implication
7 Jan 28 Predicate Logic: implications 1.5 Implication
8 Jan 30 Introduction to proofs for universal statements 2.1 Proofs for universal statements
9 Feb 2 Continuing on proofs for universal statements; existential statements 2.1-2.2 Notes from Lecture 8
Existential statements
10 Feb 4 Common mistakes in proofs 2.1-2.2 Common mistakes
Slides
11 Feb 6 Mathematical induction 2.3 Side note on divisibility
Weak induction
12 Feb 9 Mathematical induction
(recapping on weak induction)
2.3 In-class exercise
13 Feb 11 Mathematical induction
(strong induction, induction proofs for programs)
2.3 Induction proofs for programs
14 Feb 13 Strong induction 2.4 See 2nd half of note from Feb 6
15 Feb 16 Proof by contradiction 2.5 Proof by contradiction
16 Feb 18 Pigeonhole principle; set definition and operations 2.5, 3.1-3.2 Pigeonhole principle
17 Feb 20 Set definition and operations 3.3 Set definition and operations
18 Feb 23 Proving set properties 3.3 Proving set properties
19 Feb 25 Functions and relations: definition 4.1 Functions and relations 1
20 Feb 27 Intro to boolean algebra and combinatorial logic circuits 3.4-3.5 Try buidling your own circuits!
21 Mar 2 Relations and functions: recap and examples 4.1 (and a bit of 4.3) Functions and relations 2
22 Mar 4 Relations and functions: proving a function is injective/surjective 4.3 Functions and relations 3
23 Mar 6 Relations and functions: composition and inverse 4.1-4.2 Functions and relations 4
24 Mar 9 Cardinality of sets: an application of injective functions 4.3 Cardinality
25 Mar 11 Equivalence relations 4.5 Equivalence relations
26 Mar 13 Growth of functions 4.8 Growth of functions
27 Mar 16 Growth of functions (continuing from prev. lecture) 4.8
28 Mar 18 Growth of functions: master theorem 4.8 Master theorem
29 Mar 20 Introduction to combinatorics: case studies, product rule, four basic structures 5.1, 5.2 Intro to combinatorics 1
30 Mar 30 Introduction to combinatorics: four basic structuresand basic counting rules 5.1, 5.2 Intro to combinatorics 2
31 Apr 1 Combinatorics: basic counting rules, tree diagrams 5.1, 5.2 In-class worksheet
32 Apr 3 Permutations and combinations 5.2, 5.3 Permutations and combinations
Solution to in-class ex. from Wed
33 Apr 6 Counting unordered lists: stars-and-bars and recursive counting 5.4, 5.5 Counting lists
34 Apr 8 Recursive counting and solving recurrences 5.5, 5.6 Recursive counting
Solving recurrences
35 Apr 10 Basic probability: introduction, mutually exclusive events, sum rule, complementary rule 6.1-6.2 Intro to probability
36 Apr 13 Basic probability: product rule, conditional probability, independent events, Bernoulli trials 6.2 Conditional probability; Bernoulli trial
37 Apr 15 Basic probability: games of chances, expectation 6.3-6.4 Some recap on Bernoulli trial
Random variables, expectation and variance
38 Apr 17 Expectation in Bernoulli trials; variance 6.6 (See prev. lecture)
39 Apr 20 Graph Theory: Basic introduction, paths, cycles, walks, tours, Eulerian tours 7.1 Basic notions in graph theory
40 Apr 22 Bipartite graphs and vertex coloring Bipartite graphs and vertex coloring
41 Apr 24 Connected components; trees See the notes from the previous lecture
Trees
42 Apr 27 Intro to cryptography Intro to Cryptography
43 Apr 29 Intro to cryptography See the notes from the previous lecture
44 May 1 Review

Course grading

In summary of the course work listed above, the final course grade consists of the following components:

  • Weekly written assignments (30% of the final grade).

  • Weekly Moodle assignments (10% of the final grade).

  • In-class quizzes (30% of the final grade).

  • Final exam (20% of the grade).

  • Participation (10% of the grade). Class participation is awarded based mainly on your class preparation, as judged by your general attendance, classroom behavior, interaction in class, willingness to answer questions in class and at the on-line forums, and demonstrating knowledge of weekly reading during problem solving time.

The instructor may make minor modifications to this breakup as the semester progresses.

Note: To obtain a passing grade, the student has to appear for the final exam and obtain a score of at least 25/100 or above.

We will omit two assignments with the lowest scores while computing the overall assignment grade, and omit one quiz with the lowest scores while computing the overall quiz grade.

The final grades will be based on the cumulative score from all these factors (out of 100).

Collaboration Policy

  • Inspiration is free: you may discuss homework assignments with anyone. You are especially encouraged to discuss solutions with your instructor and your classmates.

  • Plagiarism is forbidden: the assignments that you turn in should be written entirely on your own. While writing the assignment you are not allowed to consult any source other than the textbook(s) for the class, your own class notes or the lecture notes for the class. Copying/consulting from the solution of another classmate constitutes a violation of the course's collaboration policy and the honor code.

  • Do not search for a solution on-line: You may not actively search for a solution to the problem from the internet. This includes posting to newsgroup or asking experts at other universities.

  • When in doubt, ask: If you have doubts about this policy or would like to discuss specific cases, please ask the instructor.

Honor Code

We will expect strict adherence to our our honor code. Please read and understand the code thoroughly. If in doubt, ask the instructor. At the end of the day, honor code violators hurt themselves by sacrificing their integrity and risking hard-earned reputation for a few measly grade points.