College of DuPage FY Fall/17
ACTIVE COURSE FILE
Curricular Area: Mathematics Course Number: 2300
Course Title: Mathematical Proof
Semester Credit Hours: 3 Lecture Hours: 3 Lab Hours: 0 Clinical Hours: 0
This course is not an IAI approved general education course.
Changes from the present course must be accompanied by a yellow Course Revision or Deletion Form.
Course description to appear in catalog:
This course serves as a transition to upper level mathematics with a focus on writing proofs. Topics include: propositional logic, predicate logic, set theory, mathematical induction, number theory, relations, and function.
Prerequisite:
Math 2233 with a grade of C or better or equivalent
A. General Course Objectives
Upon successful completion of this course students should be able to
do the following:
1. Identify propositions and differentiate propositions from non-propositions
2. Translate propositions written in English to symbolic logic form
3. Translate propositions written in symbolic logic to proper English
4. Construct truth tables for propositional forms
5. Determine if two propositional forms are logically equivalent using truth tables
6. Use truth tables to show a logical argument to be valid or invalid
7. Create two-column proofs using rules of inference and replacement
8. Perform substitutions from various domains into a predicate
9. Determine whether sentences with quantifiers are true or false
10. Construct the negation of a sentence with quantifiers
11. Develop two-column proofs using predicate logic
12. Use the structure of two-column proofs to formulate elementary paragraph proofs using standard methods such as direct or indirect proof
13. Construct proofs for bi-conditional statements
14. Construct other types of proofs including disjunctions and cases
15. Describe sets using both roster and set –builder forms
16. Demonstrate that one set is a subset of another
17. Give an example to demonstrate that one set is not a subset of another
18. Construct a proof for the equality of two sets
19. Use an index set to describe a family of sets
20. Use mathematical induction in proofs for sentences quantified over the natural numbers
21. Construct proofs using the second form of mathematical induction
22. Demonstrate basic results in number theory concerning divisibility, primes, and congruence
23. Create proofs for basic results about relations
24. Construct proofs on whether a relation is an equivalence relation
25. Differentiate between relations that are functions and those that are not functions
26. Construct proofs using the function operations
27. Demonstrate whether a function is one-to-one or onto
B. Topical Outline:
1. Propositional logic
a. Propositions
b. Propositional forms
c. Truth tables
d. Valid and invalid arguments
e. Rules of inference
f. Rules of replacement
2. Predicate logic
a. Predicates and domain
b. Substitutions
c. Quantification
d. Quantifier negation
e. Proofs involving quantified propositions
3. Proof methods
a. Direct proof
b. Indirect proof
c. Bi-conditional proof
d. Proof of disjunctions
e. Proof by cases
4. Set theory
a. Fundamentals
b. Rosters and set-builder form
c. Subsets
d. Equality of sets
e. Families of sets
f. Generalized union and intersection (optional)
5. Mathematical induction
a. The first principle
b. The second principle
c. The well-ordering principle (optional)
6. Number theory
a. Axioms (optional)
b. Divisibility
c. Primes
d. Congruence
7. Relations and functions
a. Relations
b. Equivalence relations
c. Functions
d. Function operations
e. One-to-one and onto
f. Images and inverse images (optional)
g. Cardinality (optional)
C. Methods of Evaluating Students:
Unit tests at appropriate intervals; quizzes, homework, projects, and a comprehensive final examination, all at the discretion of the instructor
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Initiator Date Division Dean Date
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Sponsor Date
Textbook for Math 2300
Title: The Structure of Proof : With Logic and Set Theory
Author: Michael O’Leary
Publisher: Prentice Hall
Copyright: 2002
The following chapters and sections of the textbook should be covered:
Chapter 1: All sections
Chapter 2: All sections (UG, UI, EG, and EI of 2.4 are optional)
Chapter 3: Sections 3.1 – 3.4 (Section 3.5 is optional)
Chapter 4: Sections 4.1 – 4.3 (Section 4.4 is optional)
Chapter 5: Sections 5.2 – 5.4 (Section 5.1 is optional)
Chapter 6: Sections 6.1 – 6.5 (Sections 6.6 and 6.7 are optional)
Use of Technology in Math 2300
The mathematics faculty recommends to all mathematics instructors that any technology be allowed and encouraged in any level mathematics course when it can be used by a student to either
1. simplify calculations where the mechanics of the problem have already been mastered or
2. explore and experiment with concepts and problems that enrich the understanding of the material that is being taught.
In all Mathematics courses, students with a documented learning disability that specifically requires a calculator as determined by Health Services, will be allowed to use a basic calculator for all test/quiz questions where arithmetic calculations are not the main objective. The specific disability must be verified with Health Services before the accommodation can be made.