Mathematical Foundations
- Course syllabus
- Mathematics 300 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:
- 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 calculus in mathematical proofs.
- 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.
- Functions and Relations: a. General definition of relations on sets. b. General features and special kinds of relations. c. Partial orders, equivalence relations. and partitions. d. Basic properties of functions. e. Common types of functions.
- 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.
- Common Methods of a Mathematical Proof a. Proof by induction b. Proof by contradiction c. Dis-proof by a counter example.