Chabot College
Fall 2013
Course Outline for Mathematics 8
DISCRETE MATHEMATICS
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Catalog Description:
MTH 8 - Discrete Mathematics
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4.00 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: MTH 1 (completed with a grade of "C" or higher)
Units
Contact Hours
Week
Term
4.00
Lecture
Laboratory
Clinical
Total
4.00
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4.00
0
0.00
4.00
70.00
0
0.00
70.00
Prerequisite Skills:
Before entry into this course, the student should be
able to:
1.
2.
3.
4.
5.
6.
7.
8.
9.
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use delta notation;
explain limits and continuity;
use Newton’s method;
apply the definition of the derivative of a function;
differentiate algebraic and trigonometric functions;
apply the chain rule;
sketch the graph of a differentiable function;
apply the Mean Value Theorem;
find the value of a definite integral as the limit of a Riemann sum;
Expected Outcomes for Students:
Upon completion of this course, the student should be
able to:
1.
2.
3.
4.
5.
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;
6.
solve recurrence relations and apply them to the analysis of
recursive programs;
7. apply concepts of graph theory to path problems (e.g., shortest path,
Euler path, Hamilton path);
8. apply properties of trees to analysis of simple games and sorting
problems;
9. apply laws of Boolean algebra to simplifying logic circuits;
10. design a finite automaton to recognize a given language.
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Course Content:
1.
2.
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 )
3. Sets, functions and relations
4. Boolean algebra, logic circuits, Karnaugh maps
5. Mathematical induction and it's relation to recursion, recurrence
equations
6. Big Oh notation, complexity of algorithms
7. Counting: permutations and combinations, inclusion?exclusion
principle, pigeonhole principle, divide and conquer algorithms
8. Graphs: Euler and Hamilton paths, coloring, isomorphism,
representations, minimal path, planarity, connectivity
9. Trees: traversal, minimal spanning trees, game trees
10. Finite automata, languages
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Methods of Presentation
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2.
3.
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Lecture/Discussion
Demonstration/Exercise
Problem Solving
Assignments and Methods of Evaluating
Student Progress
1. Typical Assignments
A.
B.
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.
Let f(x) = x^2 +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.
2. Methods of Evaluating Student Progress
A.
B.
C.
D.
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Textbook (Typical):
1.
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Exams/Tests
Quizzes
Home Work
Final Examination
Rosen, Kenneth (2012). Discrete Mathematics (7th/e). McGraw-Hill
Publishers.
Special Student Materials
1.
A calculator may be required.