CSC 202 - Discrete Mathematics for Computer Scientists II

Shai Simonson    306 Stanger    (508) 565-1008

Email:  shai@stonehill.edu

Homepage: http://www.stonehill.edu/compsci/shai.htm


Lectures:  MW 2:30 - 2:45,  001 Stanger

Texts: Discrete Mathematics and its Applications, Rosen, McGraw Hill, 7th edition.

Pettofrezzo, Anthony J. Matrices and Transformations. New York: Dover, 1978.
Exams:  There will be 6 quizzes (25%), the lowest grade will be dropped, and one final examination (35%).  The final examination will be on Tuesday, May 9,  11 AM.

Teaching Assistant:    Olivia Roy  There will be help sessions every Monday night 7 PM in the lab, or by appointment.

Assignments:  Homeworks are worth 40% of your grade.   You may do these with a partner, and one grade will be given to both people in the group.  Read our department's academic integrity guidelines before you hand in any written work.

Goals:  To understand the mathematics that underlies computer science, and to appreciate where it is used.  Last semester concentrated on functions, number theory, recurrence equations, recursion, combinatorics, and their applications.  This semester concentrates on sets, graphs, Boolean algebra, linear algebra, and their applications.

Special Dates:  None

Description: The two semester discrete math sequence covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, linear algebra, and number theory. Emphasis will be placed on providing a context for the application of the mathematics within computer science. The analysis of algorithms requires the ability to count the number of operations in an algorithm. Recursive algorithms in particular depend on the solution to a recurrence equation, and a proof of correctness by mathematical induction. The design of a digital circuit requires the knowledge of Boolean algebra. Software engineering uses sets, graphs, trees and other data structures. Number theory is at the heart of secure messaging systems and cryptography. Logic is used in AI research in theorem proving and in database query systems. Proofs by induction and the more general notions of mathematical proof are ubiquitous in theory of computation, compiler design and formal grammars. Probabilistic notions crop up in architectural trade-offs in hardware design.  Linear algebra has a vast variety of applications including: Markov chains, cryptography,  computer graphics, curve fitting, electrical circuits, and data mining.  The first semester concentrates on induction, proofs, combinatorics, recurrence relations, computational complexity, Big-O, and number theory.  The second semester concentrates on logic, sets, set algebra, countability, functions, Boolean algebra, linear algebra and applications.

 

Useful Links

Web Mining Using Eigenvalues
      The Paper
Math at Google Math in CS Halting Problem Video
Monopoly Markov Chain Matrix Calculator Wolfram Alpha My Video Lectures

Assignments

Readings:
 
Assignment 1
(Due Tuesday, Feb. 14)
                                             Assignment_2
(Due Friday after Spring Break)                         
Assignment 3
(Due before Easter Break)
Assignment 4
(Due a week before Last Day of Class)
Assignment 5
(Due before Final)

 

Brief Syllabus

Week Topics Reading
1-2 Set Theory - Inclusion/Exclusion Theorem.
Set Algebra:  Associative, Distributive, and De Morgan's Laws.  Applications of sets:
Bit Operations in Java, Union-Find data structure, functions, 1-1 correspondence, and Countability -  Diagonalization. 
and applications to Computability and Undecidability.
Rosen:  2.1 - 2.5, 8.5
3-4
Boolean Algebra.  Truth tables. Applications to Propositional and First Order Logic -  Predicates, Quantifiers, Formal proofs of Set Theorems 
Applications to automatic theorem proving, and Resolution.  Prolog and AI.
EightQueens-PrologExample
Rosen:  1.1 - 1.5
5-6
Boolean Algebra - NP-Complete Theory and Reductions, Satisfiability, 3SAT and 2SAT, operators, completeness, normal forms, identities.
Karnaugh maps, applications to circuits.
Rosen:  12.1 - 12.4
7
Linear Algebra - Introduction.  Matrices, addition, multiplication, and motivation. 
Associative, distributive laws.
Rosen:  2.6, 
Pettofrezzo: 1.1-1.4 
8
LA - Theory of Solving Equations, Gaussian elimination, Diagonal and triangular matrices. Petto: 2.5.
9
LA - Inverses and Gauss/Jordan elimination, linear independence, bases. Petto:  2.2, 2.4
10
LA - Determinants for n by n matrices, properties of determinants and the relationship to inverses. 
Transpose and theorems.
Petto:  2.1
11
LA - Geometric interpretations, and linear transformations.  Applications: Computer graphics. Petto:  3.1-3.5
12 LA - Eigenvalues and Eigenvectors, Diagonalizing a Matrix, Similar Matrices.
The Hamilton-Caley Theorem and calculating powers of a matrix. 
Petto:  4.1-4.5
13 LA - Applications -  Linear Regression, Least Squares, Curve Fitting. Cryptography and Matrix Ciphers.  Class Notes
14 LA - Applications - Probablility and Markov Chains.  Class Notes
15 LA - More Applications if time allows - More Web Mining, Warps and Morphs and More Computer Graphics. Class Notes