Download One major part of the course focuses on learning to write logically

255
Math 2405
Discrete Math
Chair: Gustavo Cepparo 223-4443 [email protected]
A full list of committee members can be found at
http://www.austincc.edu/mthdept5/mman09/cdocs/coursecommittees
Notes for Instructors
(2009-2010)
Text: Discrete Mathematics with Applications, 3rd edition, by Susanna S. Epp, Thomson
(Brooks/Cole), 2006, ISBN 0-534-35945-0
The main objective of Math 2405 is to prepare math and computer science majors for a
background in abstraction, notation and critical thinking of discrete mathematics, by
covering the basics of mathematical reasoning and problem solving. One major part of
the course focuses on learning to write logically sound mathematical arguments and to
analyze such arguments. Topics to be covered include various proof techniques, formal
logic, mathematical induction, lists, sets, relations, functions, probability and graph
theory. Students who enroll in this course are majoring primarily in mathematics,
computer science, engineering, planning to transfer theses credits to a four-year
institution.
Prerequisites: The prerequisite for discrete math is completion of Calculus I, Math 2413
or Business Calculus and Applications, Math 1425. It is not uncommon for students to
be simultaneously enrolled in Calculus II and Discrete Mathematics.
Syllabus: The course covers sections in the following order; 1.1-1.4, 2.1-2.4, 3.1-3.4, 3.6,
3.7, 5.1, 10.1, 10.2, 10.3, 6.1-6.4, 6.8, 6.9, 7.1, 7.2, 9.1, 9.2, 4.1, 4.2, 8.1, 8.2, 11.1
Chapter 1: logical form and logical equivalence, conditional statements, valid and invalid
arguments, digital logic circuits.
Chapter 2: introduction to predicates and quantified statements, multiple quantifiers and
arguments with quantifiers.
Chapter 3: direct proof and counterexample with existential and universal statements,
with rational numbers, with divisibility, with division into cases.
Chapter 5: basic definitions of set theory.
Chapter 10: relations on sets, reflexivity, symmetry and transitivity, equivalence
relations.
Chapter 6: counting and discrete probability, expected value, conditional probability,
Bayes’ theorem, independent events.
Chapter 7: functions defined on general sets, one-to-one, onto, inverse functions.
Chapter 9: real valued functions, big-O, big-omega, big-theta.
Chapter 4: sequences and mathematical induction.
Chapter 8: recursively defined sequences, solving recurrence relation by iteration.
Chapter 11: an introduction to graphs.
Exams and Grading: Exams should check student understanding on a broad front. Plan to
include questions regarding: definitions, computational problems, and proofs similar to
those discussed in class and assigned on homework. Proofs are initially difficult for
beginning students, but they make remarkable improvement later in the course.
Homework should be graded. One plan is to collect homework at each exam and grade a
few selected problems that count as 10% of the exam grade. You should give at least 4
exams.
256
Attendance: You should keep track of attendance and you may drop students who miss
more than four classes. Be sure students have in writing on the first class day that you
might drop them for more than four absences. Some students who stop attending expect
their instructor to fill out a withdrawal form for them. Your first day handouts should
indicate that you will not be responsible for withdrawing students. In general, require
students to take care of their own paperwork. You should announce your policy in
writing on your first-day handout.
257
Discrete Mathematics
First Day Handout for Students
[Semester]
MATH 2405 - [section number]
[Instructor Name]
Synonym: [insert]
[Instructor ACC Phone]
[Time], [Campus] [Room]
[Instructor email]
[Instructor web page, if applicable]
[Instructor Office]
Office Hours: [day, time]
Other hours by appointment
COURSE DESCRIPTION
MATH 2405 DISCRETE MATHEMATICS (4-4-0). A course designed to prepare math,
computer science and engineering majors for a background in abstraction, notation and critical
thinking for the mathematics 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, elementary number theory and graph theory. Skills: S Prerequisites: MATH 1425 or
MATH 2413 with C or better. ( ) Course Type: T
REQUIRED TEXTS/MATERIALS
The required textbook for this course is:
Text: Discrete Mathematics with Applications, 3rd edition, by Susanna S. Epp, Thomson
(Brooks/Cole), 2006, ISBN 0-534-35945-0
Calculators
The use of calculators or computers in order to perform routine computations is encouraged in
order to give students more time on abstract concepts. Most ACC faculty are familiar with the TI
family of graphing calculators. Hence, TI calculators are highly recommended for student use.
Other calculator brands can also be used. Your instructor will determine the
extent of calculator use in your class section.
INSTRUCTIONAL METHODOLOGY
This course is taught in the classroom as a lecture/discussion course.
COURSE RATIONALE
One major part of the course focuses on learning to write logically sound mathematical arguments
and to analyze such arguments. Students who enroll in this course are majoring primarily in
mathematics, computer science, engineering, planning to transfer theses credits to a four-year
institution.
258
COMMON COURSE OBJECTIVES
Course Measurable Learning Objectives:
Upon completion of this course students should be able to do the following:
1. Discuss definitions and diagram strategies for potential proofs in logical sequential order
without mathematical symbols (plain English).
2. Construct mathematical arguments using logical connectives and quantifiers.
3. Verify the correctness of an argument using symbolic logic and truth tables.
4. Construct proofs using direct proof, proof by contradiction, and proof by cases, or
mathematical induction.
5. Solve problems using counting techniques and combinatorics.
6. Perform operations on discrete structures such as sets, functions, relations or sequences.
7. Solve problems involving recurrence relations and generating functions.
8. Construct functions and apply counting techniques on sets in the context of discrete
probability
9. Apply algorithms and use definitions to solve problems to proof statements in elementary
number theory.
10. Use graphs and trees as a tool to visualize and simplify situations.
The topics that will enable this course to meet its objectives are:
The course covers sections in the following order; 1.1-1.4, 2.1-2.4, 3.1-3.4, 3.6, 3.7, 5.1, 10.1,
10.2, 10.3, 6.1-6.4, 6.8, 6.9, 7.1, 7.2, 9.1, 9.2, 4.1, 4.2, 8.1, 8.2, 11.1
Chapter 1: logical form and logical equivalence, conditional statements, valid and invalid
arguments, digital logic circuits.
Chapter 2: introduction to predicates and quantified statements, multiple quantifiers and
arguments with quantifiers.
Chapter 3: direct proof and counterexample with existential and universal statements, with
rational numbers, with divisibility, with division into cases.
Chapter 5: basic definitions of set theory.
Chapter 10: relations on sets, reflexivity, symmetry and transitivity, equivalence relations.
Chapter 6: counting and discrete probability, expected value, conditional probability, Bayes’
theorem, independent events.
Chapter 7: functions defined on general sets, one-to-one, onto, inverse functions.
Chapter 9: real valued functions, big-O, big-omega, big-theta.
Chapter 4: sequences and mathematical induction.
Chapter 8: recursively defined sequences, solving recurrence relation by iteration.
Chapter 11: introduction to graph theory.
259
COURSE EVALUATION/GRADING SCHEME
Grading criteria must be clearly explained in the syllabus. The criteria should specify the number
of exams and other graded material (homework, assignments, projects, etc.). Instructors should
discuss the format and administration of exams Guidelines for other graded materials, such as
homework or projects, should also be included in the syllabus.
The following policies are listed in First Day Handout section in front part of the Math Manual or
on website at http://www2.austincc.edu/mthdept5/mman09/statements.html. Insert the full
statement for each of the following in your syllabus:
Statement on Scholastic Dishonesty
Recommended Statement on Scholastic Dishonesty Penalty
Recommended Statement on Student Discipline
Statement on Students with Disabilities
Statement on Academic Freedom
COURSE POLICIES
The syllabus should contain the following policies of the instructor:
 missed exam policy
 policy about late work (if applicable)
 class participation expectations
 reinstatement policy (if applicable)
student discipline
Attendance Policy (if no attendance policy, students must be told that)
The recommended attendance policy follows. Instructors who have a different policy are required
to state it.
Attendance is required in this course. Students who miss more than 4 classes may be withdrawn.
Withdrawal Policy (including the withdrawal deadline for the semester)
It is the student's responsibility to initiate all withdrawals in this course. The instructor may
withdraw students for excessive absences (4) but makes no commitment to do this for the student.
After the withdrawal date, neither the student nor the instructor may initiate a withdrawal.
Incomplete Grade Policy
Incomplete grades (I) will be given only in very rare circumstances. Generally, to receive a grade
of "I", a student must have taken all examinations, be passing, and after the last date to withdraw,
have a personal tragedy occur which prevents course completion.
Course-Specific Support Services
ACC main campuses have Learning Labs which offer free first-come first-serve tutoring in
mathematics courses. The locations, contact information and hours of availability of the Learning
Labs are posted at: http://www.austincc.edu/tutor
260
COURSE CALENDAR/OUTLINE
16-Week Semester
Week Sections
1
1.1, 1.2
2
1.3, 1.4
3
2.1, 2.2
4
2.3, 2.4
5
3.1, 3.2
6
3.3, 3.4
7
3.6, 3.7
8
5.1, 10.1, 10.2
9
10.3, 6.1, 6.2
10
6.3, 6.4
11
6.8, 6.9
12
7.1, 7.2
13
9.1, 9.2
14
4.1, 4.2
15
8.1, 8.2
16
11.1 Review, Final Test
Instructors are encouraged to add a statement of variance, such as “Please note: schedule
changes may occur during the semester. Any changes will be announced in class.”
TESTING CENTER POLICY
ACC Testing Center policies can be found at: http://www.austincc.edu/testctr/
Instructor will add any personal policy on the use of the testing center.
STUDENT SERVICES
The web address for student services is: http://www.austincc.edu/support
The ACC student handbook can be found at: http://www.austincc.edu/handbook