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SYLLABUS
DISCRETE MATHEMATICS
Spring 2004
INSTRUCTOR:
Dr. Tomas Kovarik
tkovatrik@raritanval.edu
tel: (908) 526-1200 ext. 8421
OFFICE HOURS: W, F 12: 30 – 1:20 Tu,Th 12:00 – 1:30, 9:00 -9:50 Plus by
appointment
TEXT:
Discrete Mathematics and Its Applications, Rosen (fifth edition)
COVERAGE: ( Please, see the attached notes for detailed coverage)
1) Logic, Set Theory and Foundations (chapters 1, 3, 7, plus lecture notes)
2) Counting and Probability (chapters 4, 5, 6 plus lecture notes)
3) Integers, Groups and Boolean Algebra (chapter 2 , 10 plus lecture notes)
4) Graph Theory and Trees(chapters 8 and 9)
5) Formal Languages and Automata (if time permits)
EVALUATION:
a) 3tests (300 points)
b) Cumulative Final exam (200 points)
c) Project/ Lab (100 points)
Also, there will be presentations included in the lectures, depending on the student’s
interests.
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LEARNING OBJECTIVES:
The student will be able to understand mathematical topics essential in computer science.
The student will be able to understand and apply the concepts of proof, recursion,
induction, modeling and algorithmic thinking. The student will be able to apply concepts
learned in this course to many areas of computer science and mathematics.
TENTATIVE WEEKLY SCHEDULES
WEEK 1 (Propositional and Predicate Logic)
Simple and Compound Propositions
Truth Tables
Negation
Conjunction
Exclusive and Inclusive Disjunction
Implication (Conditionals)
Necessary and Sufficient Conditions
Inversion and Conversion of a Conditional statement
Contrapositive of a Conditional statement
Equivalent Statements (Biconditionals)
Tautology , Contradiction and Contingencies
Logical Equivalence
DeMorgan’s Laws
List of basic equivalencies
Negating the conditional and biconditional
Expressing conditional and biconditional in terms of conjunction and disjunction
Principle of Duality
Proofs of important logic rules (algebraic derivations)
Rules of precedence of logical operators
Translating from English to Logic and back
3
Predicate (propositional) functions
Universal and Existential Quantifiers
Negation of quantified statements
Nested Quantifiers
Negation of nested quantifiers
WEEK 2 (Logic of Arguments, Digital Circuits)
Validity of Arguments
Modus Ponens and Tollens
Fallacies
Error by Inversion and Conversion
Hypothetical and Disjunctive Syllogism
Disjunctive Addition (generalization)
Conjunctive Simplification (particularization)
Contradiction Rule
Arguments involving Quantifiers
Series and Parallel circuits (conjunction and disjunction)
NOT-gate, AND-gate, OR-gate, multiple input gates
Input – Output table
Boolean Expression for a given Circuit
Constructing a Circuit for a given Input-Output Table using Recognizers
Simplification of Circuits using Logical Operator Algebra
WEEK 3 (Set Theory and Functions)
Universal set and Empty set
Complements and Difference, Symmetric Difference
Intersections and Unions
Venn Diagrams
Disjoint Sets
Cartesian Products and Binary Relations
Algebra of sets and set identities including DeMorgan laws
4
Isomorphism between Sets and Logic
Principle of Duality for Set
Definition of Function as a Binary Relation
Domain, Codomain, Range
Surjection, Injection and Bijection (one-to-one correspondence)
Compositions and Inverse functions
Image and inverse image of a set
Characteristic function
Floor and Ceiling function
WEEK 4 (Test 1)
WEEK 5 (Number Theory)
Inductive Definition of Natural Numbers
Definition of Integers
Properties of Division
The Well ordering principle
Euclid’s Division Algorithm
Definition of Prime Numbers
Prime Decomposition (Fundamental Theorem of Arithmetic)
Infinitude of Primes (Euclid’s proof)
Sieve of Eratosthenes and Distribution of Primes
Mersenne and Twin primes
Prime Number Theorem (discussion, not proof)
Greatest Common Divisor (gcd) and properties
Least Common Multiple (lcm) and properties
Euclid’s Algorithm for finding gcd
Euclid’s Algorighm for finding gcd (a,b) as a linear combination of a and
WEEK 6 (Modular Arithmetic)
Congruence as an Equivalence Relation
5
Cancellation law for Congruences
Adding and Multiplying Congruences
Divisibility tests for 9 and 11
Application to Cryptology
Binary, Ternary, Hexadecimal, Octal and other Number Systems
Linear Congruences (solving them) and Inverse Modulo m
Chinese Reminder Theorem and Applications
Fermat’s Little Theorem
Pythagorean Numbers and Fermat’s Last Theorem
WEEK 7 (Mathematical Induction, Infinite Sets)
Version 1 Induction (membership property)
Version 2 Induction (general property, starting with 1)
Version 3 Induction (shifted version, starting with any Z)
Version 4 Induction (Strong Induction Principle)
Proof of Induction Principle using the Well Ordering Axiom
Equipotence of Sets as an Equivalence Relation
Cardinality and Power of a set
Countable vs. Uncountable sets
Cantor’s Diagonal Process
Aleph Zero vs. Continuum
Transfinite Arithmetic
Inclusion – Exclusion principle for finite sets
Number of Subsets in a Finite Set (Induction Proof)
Hierarchy of Cardinal Numbers
Riemann Continuum hypothesis.
WEEK 8 (Review and Test 2)
WEEK 9 (Principles of Counting)
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Basic Counting Rule
Tree Diagrams
Principle of Inclusion- Exclusion
The Pigeonhole Principle
Permutations with and without repetition
Combinations with and without repetition
Partitions and Multinomial Coefficient
Pascal’s Triangle and Binomial Coefficients
Binomial and Multinomial Theorem
Counting the number of subsets of a set with n elements
WEEK 10 (Discrete Probability Theory)
Classical Definition
Axiomatic approach (Kolmogorov)
Conditional probability
Discrete Random Variables
Binomial and Geometric Variables
Expectation and Variance
WEEK 11 (Binary Relations and Boolean Algebras)
Representing relations using graphs and matrices
Equivalence Relations
Partial and Total Orders
Closures
Boolean Functions and algebras
Aplications
WEEK 12 (Review and test 3)
WEEK 13 (Graphs and Trees)
Graphs and Isomorphisms, Konigsberg problem
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Euler and Hamilton’s paths
Shortest distance problems
Spanning Trees
Minimal Spanning Trees
WEEK 14 (Preparation and Final Exam)