## Mathematics 215

### Mathematical Proof

Length of Course: 14 weeks

Classroom Hours per Week: 4 hours

Number of Credits: 3 credits

Prerequisites: Math 114

### Course Description

A second year course for students majoring in mathematics and science. The emphasis is on understanding different proof techniques in mathematics and writing correct and clear proofs.

### Course Outline

Week Topics Covered
1 Sets (Describing a set, Subsets, Set operations, Indexed collections of sets, Partitions of sets, Cartesian products of sets)
2 Logic (Statements, The negations of a statement, The disjunction and conjunction of statements, The implication, The Biconditional, Tautologies and Contradictions, Logical equivalence, Quantified statements, Characterizations of statements)
3 Direct Proofs and Proof by Contrapositive (Trivial and vacuous proofs, Direct proofs, Proof by contrapositive, Proof by cases, Proof evaluations)
4 Proofs involving divisibility of integers, Proofs involving congruence of integers, Proofs involving real numbers, Proofs involving sets
5 Fundamental properties of set operations, Proofs involving Cartesian product of sets.
6 Existence and Proof by Contradiction (A review of three proof techniques, Existence proofs, Disproving existence statements)
7 Mathematical Induction (The principle of mathematical induction, a more general principle of mathematical induction, proof by minimum counterexample, the strong principle of mathematical induction)
8 Equivalence Relations (Relations, Properties of relations, Equivalence relations, Properties of equivalence classes)
9 Equivalence Relations (Congruence module n, the integers modulo n).
Functions (The definition of function, The set of all functions from A to B, Injective functions, Surjective functions, Bijective functions)
10 Functions (Composition of Functions, Inverse functions, Permutations).
Cardinality of sets (Numerically equivalent sets)
11 Cardinality of sets (Denumerable sets, Uncountable sets, Comparing cardinalities of sets)
12 Cardinality of sets (The Schroder-Bernstein theorem)
13 Review
14 Final Exam

### Evaluation

20% Ten Quizzes Midterm Exam Final Exam

### Textbook

Mathematical Proofs: A transition to Advanced Mathematics by Chartrand, Polimeni, Zhang, Third Edition

### Transferability

Please refer to the BC Transfer Guide www.bctransferguide.ca

### Instructors

Hayri Ardal, B.Sc.(Bogazici), Ph.D.(Simon Fraser)
Kim Peu Chew, B.Sc. (Nanjing), M.A., Ph.D. (British Columbia)