This is a course designed for math majors and others interested in higher mathematics who have completed (or will complete during the same semester) the linear algebra course Math 340 but may need a little more preparation before taking the 400 or 500 level math courses. It is specially recommended for students in the Education department.
The course serves two purposes: 1. It will teach the writing of rigorous mathematical proofs. It will cover some basic concepts of logic needed for mathematical proofs and then work with them on many examples from different areas of mathematics. 2. The course also will introduce the student to some of the abstract concepts of mathematics used in all the 400 and 500 level math courses, such as equivalence relations, orderings, and a rigorous treatment of mathematical induction and the real numbers. The 400 and 500 level math courses will assume students to be somewhat familiar with these concepts, and this course will thus give students a head start. It is specially recommended for math majors in the School of Education. Math majors in Letters and Science might want to consider Math 341 or Math 421 for an introduction to proof-making before they take 5XX-level courses.
Math 341 and Math 421 also introduce students to the writing of proofs
Higher level math courses
- Propositional logic: informal definition of formulas, truth tables, negation rules for formulas, possibly disjunctive and conjunctive normal forms and minimal sets of connectives.
- First-order predicate logic: informal definition of formulas, negation rules for formulas, possibly prenex normal form.
- Proofs: arguing by contradiction, converse, etc., proof by induction.
- Sets: set operations, subset and equality, de Morgan's laws, infinite unions and intersections, power set.
- Functions: domain/range, restriction/extension, composition, injection, surjection, bijection, infinite cartesian product, possibly informal notion of isomorphism.
- Relations: ordered pairs and tuples, finite cartesian product, notions of reflexive, symmetric, transitive.
- Equivalence relations: equivalence class, representative, partition, quotient set.
- Orderings: partial and total orderings, least and minimal element, lower bound and infimum.
- (Optional material) Analysis: rigorous linear & continuity proofs, possibly Dedekind cuts. Algebra: quotients of the ring of integers. Logic: cardinality, possibly axiom of choice.