CMSC 603, Spring '99: Brief Lecture Notes
Lecture 1: Jan 29
Course Objectives
Three parts: proving theorems, counting, modeling. Related objectives: Solving problems, calculations (using Maple etc.), research applications.
Course Structure
Homework Three assigned problems per week; additional problem solving recommended. A grader will be assigned per problem. Model solutions will be written up by assigned grader and will be verified by Dr Sherman.
Readings
Biggs (course text book); discrete math section from 641 text; The Art of Computer Programming: Vol 1 by Knuth - chapter on fundamentals.
Exams
Three exams; take home
Research Project
Proof Techniques
- Direct: by definition; implication; core analysis; construction
- Counter-example
- Indirect (contradiction)
- Counting
- Diagonalization
- Reduction
Analyzed proof of the "All Horses are Red" theorem. Error in the inductive step; cannot consider three distinct horses for n=2. Proved the following theorem: The sum of any two even integers is even.
Construction of discrete math starting from set theory. Constructed the set of natural numbers. Well ordering principle: Every non-empty subset of natural numbers has a least element.
Discussed planar graphs. Excercise: Prove V+R-E relationship for planar graphs.