Chabot College
Fall 2008
Course Outline for Mathematics 8
DISCRETE MATHEMATICS
Catalog Description:
8 - Discrete Mathematics
4 units
Sets, relations and functions; logic, methods of proof, induction; combinatorics, recursion, recurrence
relations and complexity of algorithms; graphs and trees; logic circuits; automata. Designed for majors in
mathematics and computer science. Prerequisite: Mathematics 1 (completed with a grade of “C” or higher).
4 hours.
[Typical contact hours: 70]
Prerequisite Skills:
Before entering the course, the student should be able to:
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use delta notation;
explain limits and continuity;
use Newton’s method;
apply the definition of the derivative of a function;
define velocity and acceleration in terms of mathematics;
differentiate algebraic and trigonometric functions;
apply the chain rule;
find all maxima, minima and points of inflection on an interval;
sketch the graph of a differentiable function;
apply implicit differentiation to solve related rate problems;
apply the Mean Value Theorem;
demonstrate an understanding of the definite integral as the limit of a Riemann sum;
demonstrate an understanding of the Fundamental Theorem of Integral Calculus;
demonstrate an understanding of differentials and their applications;
integrate using the substitution method;
find the volume of a solid of revolution using the shell, disc, washer methods;
find the volume of a solid by slicing;
find the work done by a force;
find the hydrostatic force on a vertical plate;
find the center of mass of a plane region;
approximate a definite integral using Simpson’s Rule and the Trapezoidal Rule.
Expected Outcomes for Students:
Upon completion of the course, the student should be able to:
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apply principles of symbolic logic to the construction of formal proofs;
prove mathematical statements using proof by contradiction, proof by contraposition and proof by
cases. (Example: prove that there are infinitely many primes.);
apply mathematical induction to problems in sequences, series, and algorithms;
solve counting problems using elementary counting techniques: sum and product rules; pigeonhole
principle, combinations and permutations; inclusion/exclusion principle;
use counting principles to measure the complexity of computer algorithms;
solve recurrence relations and apply them to the analysis of recursive programs;
apply concepts of graph theory to path problems (e.g., shortest path, Euler path, Hamilton path);
apply properties of trees to analysis of simple games and sorting problems;
apply laws of Boolean algebra to simplifying logic circuits;
design a finite automaton to recognize a given language.
Chabot College
Course Outline for Mathematics 8, Page 2
Fall 2008
Course Content:
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Symbolic logic and rules of inference
Informal proof techniques: proof by cases, proof by contradiction, proof by contraposition, existence vs.
constructive proofs. Applications in number theory (e.g., infinitude of primes, irrationality of 2 )
Sets, functions and relations
Boolean algebra, logic circuits, Karnaugh maps
Mathematical induction and it's relation to recursion, recurrence equations
Big Oh notation, complexity of algorithms
Counting: permutations and combinations, inclusion-exclusion principle, pigeonhole principle, divide
and conquer algorithms
Graphs: Euler and Hamilton paths, coloring, isomorphism, representations, minimal path, planarity,
connectivity
Trees: traversal, minimal spanning trees, game trees
Finite automata, languages
Methods of Presentation:
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Lecture/demonstration.
Discussion.
Typical Assignments and Methods of Evaluating Student Progress:
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Typical Assignments
a. How many different functions are there from a set of 6 elements to itself? How many of them are:
(a) onto? (b) not onto? (c) one-to-one? (d) not one-to-one? Design an algorithm that determines
whether a function from a set of n elements to itself is one-to-one, and another that determines
whether the function is onto.
b. Let f(x) = x2 +1, x is real on [ -2, 4]. Define a relation R on A X A as: (a, b) is in R if and only if f(a)
= f(b). Show R is an equivalence relation. Describe the equivalence classes.
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Methods of Evaluating Student Progress
a. Homework
b. Quizzes
c. Exams and final exam
Textbook(s) (Typical):
Discrete Mathematics, 6th Edition, Kenneth Rosen: McGraw-Hill Publishers, 2007
Discrete Mathematics with Applications, 3rd Edition, Suzanna Epp: Brooks/Cole Publishing Company, 2007
Special Student Materials:
A calculator may be required.
Revised 9/18/2007 J. Traugott, M. Ho