Foothill CollegeApproved Course Outlines

Physical Sciences, Mathematics & Engineering Division
5 hours lecture.5 Units

Total Quarter Learning Hours: 60 (Total of All Lecture, Lecture/Lab, and Lab hours X 12)
 Lecture Hours: 5 Lab Hours: Lecture/Lab:
 Note: If Lab hours are specified, see item 10. Lab Content below.

Repeatability -
Statement: Not Repeatable.

Status -
 Course Status: ActiveGrading: Letter Grade with P/NP option
 Degree Status: ApplicableCredit Status: Credit
 Degree or Certificate Requirement: AS Degree,   Foothill GE
 GE Status: Communication & Analytical Thinking

Articulation Office Information -
 Transferability: BothValidation: 2/3/2012; 11/14/12

Cross Listed as:MATH 22
Related ID:

1. Description -
Discrete mathematics: set theory, logic, Boolean algebra, methods of proof, mathematical induction, number theory, discrete probability, combinatorics, functions, relations, recursion, algorithm efficiencies, graphs, trees.
Prerequisite: C S 1A or 1AH; satisfactory score on the mathematics placement test or MATH 49 or 48C.
Co-requisite: None
Advisory: Eligibility for one of the following: ENGL 1A, 1AH, 1S & 1T or ESLL 26.; not open to students with credit in CIS 18 or MATH 22.

2. Course Objectives -
The student will be able to:
  1. Use formal logic in constructing valid arguments.
  2. Write proofs formally, including writing proofs using symbolic logic and Boolean Algebra.
  3. Use number theory to solve to solve problems.
  4. Understand the basics of set theory, including solving problems in combinatorics and probability theory.
  5. Prove combination and permutation principles and use them to solve problems.
  6. Understand the definition of functions.
  7. Use recursive thinking and method to solve recurrence relations, including using recursion to analyze algorithms and programs.
  8. Analyze and write algorithms.
  9. Identify relations and their properties.
  10. Draw and analyze graphs and trees, including applying matrices to analyze graphs and trees
  11. Solve Application problems from Computer Science, including using finite state machines to model computer operations.
  12. Discuss mathematical problems and write solutions in accurate mathematical language and notation.
  13. Interpret mathematical solutions.
3. Special Facilities and/or Equipment -
  1. Scientific calculator
  2. When taught hybrid: Four lecture hours per week in face-to-face contact and one hour per week using CCC Confer. Students need internet access.

4. Course Content (Body of knowledge) -
  1. Logic
    1. Logical Forms and Equivalences
    2. Conditional Statements
    3. Valid and Invalid Arguments
    4. Predicates and Quantified Statements
    5. Boolean Algebra
    6. Application: Digital Logic Circuits
  2. Methods of Proof/Proof Techniques
    1. Direct Proof
    2. Proof by Counterexample
    3. Proof by Division into Cases
    4. Proof by Contradiction and Contraposition
    5. Proof by Induction
    6. Strong Mathematical Induction and Well-Ordering
  3. Number Theory
    1. Properties of Prime and Rational Numbers
    2. Unique Factorization Theorem
    3. Quotient-Remainder Theorem
    4. Modular Arithmetic
    5. Floor and Ceiling Notation
    6. Applications of Number Theory to Problem Solving
    7. Principal of inclusion and exclusion
  4. Set Theory
    1. Notation
    2. Operations on Sets
    3. Cartesian Products
    4. Proving Set Identities
  5. Counting and Probability
    1. Events and Sample Space
    2. Possibility Trees and Multiplication Rule
    3. Addition Rule
    4. Pigeonhole Principle
    5. Combinations and Permutations
    6. Pascal's Formula
    7. Binomial Theorem
    8. Discrete Probability Axioms
    9. Expected Value
    10. Conditional Probability, Baye's Formula
    11. Integer Random variables
    12. Expectations
    13. Law of large numbers
  6. Functions
    1. One-to-One, Onto, Inverses
    2. Compositions
    3. Well Defined Functions
  7. Recursion
    1. Recursively Defined Sequences
      1. Fibonacci numbers
    2. Solving Recurrence Relations by Iteration
    3. Solving Recurrence Relations using Logarithm
    4. Verifying Solutions by Mathematical Induction
  8. Efficiency of Algorithms
    1. Big-O, Big-Theta, and Big-Omega Notation
    2. Exponential and Logarithmic Orders
    3. Computing Orders of Algorithms
    4. Analysis of Various Sort and Search Algorithms
  9. Relations
    1. Binary Relations, N-ary relations
    2. Directed Graphs
    3. Inverse Relations
    4. Reflexivity, Symmetry, and Transitivity
    5. Equivalence Relations and Classes
  10. Graphs and Trees
    1. Definitions and Properties
    2. Paths and Circuits
      1. Euler Path
      2. Hamiltonian Circuit
      3. shortest path and minimal spanning tree
    3. Matrix Representation of Graphs
    4. Isomorphisms of Graphs
    5. Spanning Trees
    6. Traversal Problems
    7. Decision trees
    8. Huffman Codes
    9. Warshall's algorithm
  11. Solve Application problems from Computer Science
    1. The application of mathematical induction to recursive computer algorithms
    2. The use of sequences in loop structures
    3. The application of computer logic
      1. AND-, OR- and NOT-GATES
      2. Boolean algebra structure
      3. logic networks
      4. minimization
    4. Breaking down problems or functions into components, sub-problems or sub-functions
    5. The use of time-complexity to determine Big-O growth rate of various algorithms
    6. Articulation points (cut vertices) and computer networks
    7. Modeling arithmetic, computation, and languages including algebraic structures, finite-state machines and formal logic
  12. Discuss mathematical problems and write solutions in accurate mathematical language and notation.
    1. Application problems from other disciplines
    2. Proper notation
  13. Interpret mathematical solutions.
    1. Explain the significance of solutions to application problems.
5. Repeatability - Moved to header area.
6. Methods of Evaluation -
  1. Written homework
  2. Quizzes, tests
  3. Proctored comprehensive final examination
7. Representative Text(s) -
Susanna S. Epp, Discrete Mathematics with Applications, 4th ed., Brooks/Cole, 2010.

8. Disciplines -
9. Method of Instruction -
Lecture, Discussion, Cooperative learning exercises.
10. Lab Content -
Not applicable.
11. Honors Description - No longer used. Integrated into main description section.
12. Types and/or Examples of Required Reading, Writing and Outside of Class Assignments -
  1. Homework Problems: Homework problems covering subject matter from text and related material ranging from 30 - 60 problems per week. Students will need to employ critical thinking in order to complete assignments
  2. Lecture: Five hours per week of lecture covering subject matter from text and related material. Reading and study of the textbook, related materials and notes
  3. Projects: Student projects covering subject matter from textbook and related materials. Projects will require students to discuss mathematical problems,write solutions in accurate mathematical language and notation and interpret mathematical solutions. Projects may require the use of a computer algebra system such as Mathematica or MATLAB
  4. Worksheets: Problems and activities covering the subject matter
Such problems and activities will require students to think critically. Such worksheets may be completed both inside and/or outside of class
13. Need/Justification -
This course is a required core course for the AS degrees in Mathematics and Computer Science.

Course status: Active
Last updated: 2015-06-25 08:04:20

Foothill CollegeApproved Course Outlines