discrete lecture 01

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CS335- Discrete Mathematics

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Agenda

Course policies

Quick Overview

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Grading Scheme

Quize + Assignments 12 Midterm 18 Final 30

Total 60

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Recommended Books

“Discrete Mathematics with Application” by Susana.

K.H. Rosen, Discrete Mathematics and its Applications, (5th Edition), McGraw Hill 1999.

“Discrete Mathematical Structures” by B. Kalman Prentice Hall (1996).

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Grading (cont’d)

Exams/Quizzes can be from the following:

Current lecture Material covered in any previous lecture Reading assignments From any assigned homework.

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Academic Dishonesty

Any form of cheating on exams/assignments/quizzes is subject to a penalty.

Assignment Copy may lead to zero in all assignments.

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Introduction

Discrete mathematics describes processes that consist of a sequence of individual steps.

This contrasts with calculus, which describes processes that change in a continuous fashion.

Whereas the ideas of calculus were fundamental to the science and technology of the industrial revolution, the ideas of discrete mathematics underlie the science and technology of the computer age

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Discrete Continuous – 5a

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What Is Discrete Mathematics?

Definition Discrete Mathematics Discrete Mathematics is a collection of

mathematical topics that examine and use finite or countably infinite mathematical objects.

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Quick Overview - Topics

Logic and Sets Make notions you’re already used to from

programming a little more rigorous (operators) Fundamental to all mathematical disciplines Useful for digital circuits, hardware design

Elementary Number Theory Get to rediscover the old reliable number and

find out some surprising facts Very useful in crypto-systems

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Quick Overview - Topics

Proofs (especially induction) If you want to debug a program beyond a doubt,

prove that it’s bug-free Proof-theory has recently also been shown to be

useful in discovering bugs in pre-production hardware

Counting and Combinatorics Compute your odds of winning lottery Important for predicting how long certain

computer program will take to finish Useful in designing algorithms

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Quick Overview - Topics

Graph Theory Many clever data-structures for organizing information

and making programs highly efficient are based on graph theory

Very useful in describing problems in Databases Operating Systems Networks EVERY CS DISCIPLINE!!!!

Trees Data structures for organizing information and making

programs efficient

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What is Discrete Mathematics – 6

Set of Integers & Real Numbers – 5b

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Logic – 7

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Statement – 8a

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Examples – 8b

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Truth Values of Propositions – 8c

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Examples – 9a

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Statements & Truth Values – 9b

TTFF

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Example – 10b

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Understanding Statements – 11c

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Example – 11b

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Compound Statement – 12a

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Symbolic Representation – 13a

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Logical Connectives – 14a

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Examples – 14b

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Translating from English to Symbols – 15

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Translating from English to Symbols – 16a

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Translating from English to Symbols – 16

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Translating from English to Symbols – 17a

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Translating from English to Symbols – 17b

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Negation – 19

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Truth Table for ~p – 20

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Conjunction – 21

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Truth Table for p ^ q – 22

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Disjunction – 23

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Truth Table

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Truth Table for ~p^q - 2

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Truth Table for ~p^q – 2a

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Truth Table for ~p^q – 2b

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Truth Table for ~p^q – 2c

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~p ^ (q v~ r) – (2 - 3a)

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~p ^ (q v~ r) – 2 - 3b

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~p ^ (q v~ r) – 2 - 3c

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~p ^ (q v~ r) – 2 - 3d

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Truth Table for ~p (p v~ q) – 2 - 3e

v

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Truth Table for (pvq) ^~ (p^q) – 2 - 4a

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Truth Table for (pvq) ^~ (p^q) – 2 - 4c

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Truth Table for (pvq) ^~ (p^q) – 2 - 4e

vv

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Truth Table for (pvq) ^~ (p^q) – 2 -4f

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Exclusive OR – 2 - 5

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Symbols for Exclusive OR – 2 - 5a

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Logical Equivalence – 2 - 6

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Double Negation ~(~p) ≡ p – 2 - 7

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Examples – 2 - 12

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Example – 2 - 17c

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Example – 2 - 17e

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De Morgan’s Laws – 2 - 9

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De Morgan’s Laws – 2 - 9a

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Proof – 2 - 16

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Proof – 2 - 16d

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Application – 2 - 10

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Exercise – 2 - 19

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Tautology – 2 - 21

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Example – 2 - 21a

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Contradiction – 2 - 22

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Example – 2 - 22a

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Exercise – 2 - 23

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Exercise – 2 - 24

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Laws of Logic – 2 - 25

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Laws of Logic – 2 - 25a

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Laws of Logic – 2 - 25b

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Laws of Logic – 2 - 25c

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Laws of Logic – 2 - 25d

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Application - 1

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Example - 2

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Simplifying a Statement – 3

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Distributive Law in Reverse – 4

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Exercise – 5

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Exercise - 5a