Discrete Mathematics - University of Colorado Boulderecee.colorado.edu/~siek/ecen3703/spring10/lecture1.pdf · Uses of Discrete Math are Everywhere I Circuit design I Computer architecture

Discrete Mathematics

Jeremy Siek

Spring 2010

Jeremy Siek Discrete Mathematics 1 / 15


Outline of Lecture 1

1. Course Information

2. Overview of Discrete Mathematics


Course Information

I Class web page:http://ecee.colorado.edu/~siek/ecen3703/spring10

I Textbooks:I Discrete Mathematics and its Applications, 6th Edition, by Rosen. (At

the CU bookstore.)I A Tutorial Introduction to Structured Isar Proofs, by Nipkow.

(Available online.)I Isabelle/HOL – A Proof Assistant for Higher-Order Logic, by Nipkow,

Paulson, and Wenzel. (Available online.)I How to Prove It: A Structured Approach, by Daniel J. Velleman.

I Grading:Quizzes 30%Midterm exam 30%Final exam 40%


Course Information: Homework

I There are weekly homework assignments.I The quizzes and exams are based on the homework.I Every students gets a personal tutor named Isabelle. Isabelle is a

logic language, a programming language, and a most importantly,a proof checker.http://www.cl.cam.ac.uk/research/hvg/Isabelle/

I You know your proofs are correct when you convince Isabelle.


Overview of Discrete Mathematics

Discrete

Mathematics


Mathematics

I What is Math anyways?

I Is it the study of numbers?I Mathematics is actually much more broad.

DefinitionMathematics is the study of any truth regarding well-definedconcepts.

Numbers are just one kind of well-defined concept.


Mathematics

I What is Math anyways?I Is it the study of numbers?

I Mathematics is actually much more broad.




Mathematics

I What is Math anyways?I Is it the study of numbers?I Mathematics is actually much more broad.




Mathematics

I What is Math anyways?I Is it the study of numbers?I Mathematics is actually much more broad.




Discrete

DefinitionSomething is discrete if is it composed of distinct, separable parts. (Incontrast to continuous.)

Discrete Continuousintegers real numbersgraphs rational numbers

state machines differential equationsdigital computer radiosquantum physics Newtonian physics


Discrete Mathematics

DefinitionDiscrete Mathematics is the study of any truth regarding discreteentities.

I That’s pretty broad. So what is it really?I Discrete math is the foundation for the rigorous understanding of

computer systems.


A Discrete Problem: Sudoku

7

1

3 9

3

1 8

6

5

6 4

3

7

7

7

7

7

2

2

2

2

2

2

3

3

3

3

1

1

4

4

45

6

6

8

8

8

9

9

9

I What are the rules of Sudoku?

I Spend the next few minutes filling in this board.I Write down the rules of Sudoku on a sheet of paper.I Pass your paper to the person on your right. Are the rules that

you’ve been passed correct? If not, give an example.



7

1

3 9

3

1 8

6

5

6 4

3

7

7

7

7

7

2

2

2

2

2

2

3

3

3

3

1

1

4

4

45

6

6

8

8

8

9

9

9

I What are the rules of Sudoku?I Spend the next few minutes filling in this board.

I Write down the rules of Sudoku on a sheet of paper.I Pass your paper to the person on your right. Are the rules that




7

1

3 9

3

1 8

6

5

6 4

3

7

7

7

7

7

2

2

2

2

2

2

3

3

3

3

1

1

4

4

45

6

6

8

8

8

9

9

9

I What are the rules of Sudoku?I Spend the next few minutes filling in this board.I Write down the rules of Sudoku on a sheet of paper.

I Pass your paper to the person on your right. Are the rules thatyou’ve been passed correct? If not, give an example.



7

1

3 9

3

1 8

6

5

6 4

3

7

7

7

7

7

2

2

2

2

2

2

3

3

3

3

1

1

4

4

45

6

6

8

8

8

9

9

9

I What are the rules of Sudoku?I Spend the next few minutes filling in this board.I Write down the rules of Sudoku on a sheet of paper.I Pass your paper to the person on your right. Are the rules that



Abstracting Sudoku

7

1

3 9

3

1 8

6

5

6 4

3

7

7

7

7

7

2

2

2

2

2

2

3

3

3

3

1

1

4

4

45

6

6

8

8

8

9

9

9

I What aspects of the game of Sudoku don’t really matter?I What could you change such that an expert Sudoku player would

immediately be an expert of the modified game?I What aspects of the game really matter?


Sudoku Solver

7

1

3 9

3

1 8

6

5

6 4

3

7

7

7

7

7

2

2

2

2

2

2

3

3

3

3

1

1

4

4

45

6

6

8

8

8

9

9

9

I Write down a pseudo-code algorithm for solving Soduku.I What data structures did you use?I What kind of algorithm did you use?I Does your algorithm always solve the puzzle?I How long does your algorithm take to finish in the worst case?


Why Study Discrete Mathematics?

I It’s the basic language used to discuss computer systems. Youneed to learn the language if you want to converse with othercomputer professionals.

I It’s a toolbox full of the problem-solving techniques that you willuse over and over in your career.

I But best of all, studying discrete math will enhance your mind,turning it into a high-precision machine!


Uses of Discrete Math are Everywhere

I Circuit designI Computer architectureI Computer networksI Operating systemsI Programming: algorithms and data structuresI Programming languagesI Computer security, encryptionI Error correcting codesI Graphics algorithms, game enginesI . . .


Themes in Discrete Math

Mathematical Reasoning: read, understand, and create precisearguments.

Discrete Structures: model discrete systems and study theirproperties.

Algorithmic Thinking: create algorithms, verify that they work,analyze their time and space requirements.

Combinatorial Analysis: counting (not always as easy as it sounds!)


Advice

I Read in advance.I Do the homework.I Form a study group.I Form an intense love/hate relationship with Isabelle.


Documents

Discrete Mathematics - University of Colorado Boulderecee.colorado.edu/~siek/ecen3703/spring10/lecture1.pdf · Uses of Discrete Math are Everywhere I Circuit design I Computer architecture