MTH202 Handouts Lectures 1-45

Table of Contents

LOGIC ..............................................................................................................................3 TRUTH TABLES.............................................................................................................9 LAWS OF LOGIC..........................................................................................................17

ARGUMENT..................................................................................................................29 APPLICATIONS OF LOGIC ........................................................................................34 SET THEORY................................................................................................................41 VENN DIAGRAM.........................................................................................................46 SET IDENTITIES ..........................................................................................................58 LECTURE NO.10 APPLICATIONS OF VENN DIAGRAM.....................................................................65 LECTURE NO.11 RELATIONS ..................................................................................................................73 LECTURE NO.12 TYPES OF RELATIONS...............................................................................................80 LECTURE NO.13 MATRIX REPRESENTATION OF RELATIONS ......................................................90 LECTURE NO.14 INVERSE OF RELATIONS..........................................................................................97 LECTURE NO.15 FUNCTIONS................................................................................................................102 LECTURE NO.16 TYPES OF FUNCTIONS ............................................................................................112 LECTURE NO.17 INVERSE FUNCTION................................................................................................123 LECTURE NO.18 COMPOSITION OF FUNCTIONS.............................................................................134 LECTURE NO.19 SEQUENCE..................................................................................................................144 LECTURE NO.20 SERIES .........................................................................................................................151 LECTURE NO.21 RECURSION I .............................................................................................................159 LECTURE NO.22 RECURSION II............................................................................................................165 LECTURE NO.23 MATHEMATICAL INDUCTION ..............................................................................170 LECTURE NO.24 MATHEMATICAL INDUCTION FOR DIVISIBILITY...........................................179 LECTURE NO.25 METHODS OF PROOF...............................................................................................186 LECTURE NO.26 PROOF BY CONTRADICTION.................................................................................193 LECTURE NO.27 ALGORITHM ..............................................................................................................201 LECTURE NO.28 DIVISION ALGORITHM ...........................................................................................205 LECTURE NO.29 COMBINATORICS.....................................................................................................210 LECTURE NO.30 PERMUTATIONS .......................................................................................................218 LECTURE NO.31 COMBINATIONS........................................................................................................226 LECTURE NO.32 K-COMBINATIONS ...................................................................................................232 LECTURE NO.33 TREE DIAGRAM ........................................................................................................238 LECTURE NO.34 INCLUSION-EXCLUSION PRINCIPLE...................................................................245 LECTURE NO.35 PROBABILITY............................................................................................................250 LECTURE NO.36 LAWS OF PROBABILITY .........................................................................................256 LECTURE NO. 37 CONDITIONAL PROBABILITY..............................................................................265 LECTURE NO. 38 RANDOM VARIABLE..............................................................................................272 LECTURE NO. 39 INTRODUCTION TO GRAPHS ...............................................................................280 LECTURE NO. 40 PATHS AND CIRCUITS ...........................................................................................289 LECTURE NO. 41 MATRIX REPRESENTATION OF GRAPHS..........................................................296 LECTURE NO. 42 ISOMORPHISM OF GRAPHS...................................................................................303 LECTURE NO. 43 PLANAR GRAPHS ....................................................................................................312 LECTURE NO. 44 TREES .........................................................................................................................319 LECTURE NO. 45 SPANNING TREES....................................................................................................327 1Logic

Logic

Course Objective:

1. Express statements with the precision of formal logic
2. Analyze arguments to test their validity
3. Apply the basic properties and operations related to sets
4. Apply to sets the basic properties and operations related to relations and functions
5. Define terms recursively
6. Prove a formula using mathematical induction
7. Prove statements using direct and indirect methods
8. Compute probability of simple and conditional events
9. Identify and use the formulas of combinatorics in different problems
10. Illustrate the basic definitions of graph theory and properties of graphs
11. Relate each major topic in Discrete Mathematics to an application area in computing

1.Recommended Books:

1. Discrete Mathematics with Applications (second edition) by Susanna S. Epp
2. Discrete Mathematics and Its Applications (fourth edition) by Kenneth H. Rosen
3. Discrete Mathematics by Ross and Wright

MAIN TOPICS:

1. Logic
2. Sets & Operations on sets
3. Relations & Their Properties
4. Functions
5. Sequences & Series
6. Recurrence Relations
7. Mathematical Induction
8. Loop Invariants
9. Loop Invariants
10. Combinatorics
11. Probability
12. Graphs and Trees

Continuous · Discrete · Set of Integers:

-2 -1

Set of Real Numbers:

-3 -2 -1

What is Discrete Mathematics?

Discrete Mathematics concerns processes that consist of a sequence of individual steps.

LOGIC:

Logic is the study of the principles and methods that distinguish between a valid and an invalid argument.

SIMPLE STATEMENT:

A statement is a declarative sentence that is either true or false but not both. A statement is also referred to as a proposition

EXAMPLES:

a. 2+2 = 4, b. It is Sunday today

If a proposition is true, we say that it has a truth value of "true”. If a proposition is false, its truth value is "false". The truth values “true” and “false” are, respectively, denoted by the letters T and F.

EXAMPLES:

Propositions

1. Grass is green.
2. 4 + 2 = 6
3. 4 + 2 = 7
4. There are four fingers in a hand.

Rule:

If the sentence is preceded by other sentences that make the pronoun or variable reference clear, then the sentence is a statement.

Example:

x = 1 x > 2 “x > 2” is a statement with truth-value FALSE.

Not Propositions

1) Close the door. 2) x is greater than 2. 3) He is very rich

Example

Bill Gates is an American He is very rich “He is very rich” is a statement with truthvalue TRUE.