Math 300: Mathematical Foundations
Prerequisites: MATH 172 and MATH 260
Credit Hours: (3)
A first course in the foundations of modern mathematics. The topics covered include sentential calculus, set theory, the number system, induction and recursion, functions and relations, and computation. The methods of proof and problem solving needed for upper-division coursework and the axiomatic basis of modern mathematics are emphasized throughout the course.
Detailed Description of Course
Course content includes:
1) Sentential Calculus:
a. Logical symbols and logical connectives.
b. Sufficient condition, necessary condition and if and only if.
c. The use of truth tables and applications.
d. Tautologies, and tautological consequences.
e Validity and satisfiability.
f. Principles for sentential calculus.
g. Using the language of predicate claculus in mathemathical proofs.
2) Fundamental Set theory:
a. Definitions of sets, subsets, elements of sets.
b. Standard notation of sets and set operations
c. Some common number sets.
d. Ordinality and cardinality.
3) Functions and Realtions:
a. General definition of realtions on sets.
b. General features and special kinds of reltaions.
c. Partial orders, equivalence relations. and partitions.
d. Basic properties of functions.
e. Common types of functions.
4) The Number System:
a. Natural Numbers, Integers, and Rational Numbers.
b. Ordinality, cardinality, and countability of Rational Numbers.
c. The Real Numbers; irrationality, and the non-denumerability of the reals.
d. The least Upper Bound and Greatest Lower Bound of a set
e. Recursion on the set of Natural Numbers.
5) Common Methods of a Mathematical Proof
a. Proof by induction
b. Proof by contradiction
c. Dis-proof by a counter example.
Detailed Description of Conduct of Course
This is a traditional lecture course, but with a significant degree of classroom interaction encouraged and collaborative (group-learning) projects, worksheets and assignments will be frequent. Students will use computers in and out of class to write their own computable functions and apply these programming techniques to solve probelms in other topics in the course.
Student Goals and Objectives of the Course
The primary objective of the course is to give students a training in writing mathematical proofs, and to prepare students for upper-division coursework in mathematics. Students will be able to:
1) Comprehend and express mathematical ideas, both orally and in writing forms, in the language of modern mathematics, including first-order logic and formal set theory.
2) Employ the most common problem solving techniques and methods of proof needed in advanced coursework.
3) Understand the axiomatic foundations of the mathematics they have previously learned, and be able to approach the study of new topics covered by Math 400 level courses using an axiomatic framework.
Graded tasks will include individual homework, quizzes, and written exams, including a cumulative final. Additional assessment measures may include collaborative projects, in-class presentations and class attendance.
Other Course Information
Review and Approval
November 7, 2017
February 20, 2017
Revised April 13, 2012