MATH 300: Mathematical Foundations

Prerequisites: MATH 142 or ITEC 122 or permission of instructor, and MATH 152 and any MATH course numbered 200 or above

Credit Hours: (3)

A first course in the foundations of modern mathematics. The topics covered include propositional and predicate logic, 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. The level of the course is challenging but appropriate for students with a minimum of 3 semesters of college mathematics. Students who have earned credit for MATH 200 may not subsequently earn credit for MATH 300.

Detailed Description of Course

Course content includes:

The propositional calculus:

- Propositional variables and logical connectives.
- The use of truth tables to test for truth conditions.
- Tautologies, and contradictions.

The predicate calculus:

- Predicate functions, variables, and logical connectives.
- The universal and existential quantifiers and their standard interpretations.
- Validity and satisfiability.
- Soundness and completeness.
- Using the language of predicate calculus in mathematical proofs.
- Naïve and formal set theory:
- Standard set notation.
- The set operations of union, intersection, symmetric difference, and power set.
- The Zermelo/Frankel axioms and the axiom of choice.
- Finite and transfinite sets, Cantor’s theorem.

Functions and Relations:

- Relations on sets, including transitive, symmetric, and reflexive relations.
- Partial orders, equivalence relations, and partitions.
- Functions on sets, including compound functions.

The Number System:

- The sets of Natural Numbers and Integers; well-foundedness and proofs by induction, ordinality and cardinality, countability, the Peano axioms.
- The Rational Numbers; rational number arithmetic and the field axioms.
- The Real Numbers; irrationality, algebraic and transcendental numbers, Dedekind cuts, and the non-denumerability of the reals.
- Other number systems; algebraic versus geometric closure of a field, extension by radicals (e.g., the Gaussian integers), transfinite ordinals and cardinals.

Computation:

- Turing machines.
- Computation and primitive functions.
- Computable functions, recursion, recursively enumerable and non-recursively enumerable sets.
- The Halting Problem.
- Church’s Thesis.

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 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 problems in other topics in the course.

Student Goals and Objectives of the Course

The primary objective of the course is to prepare students for upper-division coursework in mathematics. Students will be able to

- Comprehend and express themselves clearly in the language of modern mathematics, including first-order logic and formal set theory.
- Employ the most common problem solving techniques and methods of proof needed in advanced coursework.
- Understand the axiomatic foundations of the mathematics they have previously learned, and be able to approach the study of new topics such as modern algebra, number theory, and analysis using an axiomatic framework and the expository cycle of “definition-theorem-proof.”

Assessment Measures

Graded tasks will include individual homework, quizzes, and written exams, including a cumulative final. Additional assessment measures may include collaborative projects or homework.

Other Course Information

Review and Approval Date

Revised: April 13, 2012