MATH 245 is what we call a “bridge course” for the math major; it bridges the gap between basic calculuational courses like Calculus and more advanced theoretical courses like Abstract Algebra or Real Analysis. Specifically, in this MATH 245 class you’ll learn — perhaps for the first time — how to read, write, and communicate mathematical arguments.

Along the way, our topic of choice will be “discrete” mathematics; this covers a lot of topics, but what they all have in common is that they deal with objects that are distinct, unconnected, and separate (like sequences of numbers, or a finite set of symbols).

### Learn about Discrete Mathematics

Up until now your mathematical experience has probably focused around Calculus, which focuses on *continuous *topics like limits, rates of change, integrals, and the study of continuous functions.

In contrast, Discrete Mathematics asks questions about *discrete* topics like sequences of numbers, finite sets, and combinatorics. Discrete Mathematics topics covered in this course include logic, set theory, relations and functions, mathematical induction and equivalent forms, recurrence relations, and counting techniques.

### Learn how to read, write, and communicate with Proofs

The discrete mathematics topics listed above will be the universe in which we practice the most important objective of the course: learning how to read, write, and communicate mathematical arguments. You’ve probably seen mathematical proofs in your previous classes, and maybe had to write a few yourself. Proofs are the language that mathematicians use to communicate witih each other, and will be the backbone of this class.

Although the *material* in this course is not particularly difficult, the expectations for communication and argument should pose a significant challenge, and we will spend a lot of time working on improving your skills in that regard. In particular, throughout the course of the semester you will work at getting better at:

- Reading and understanding mathematical material on your own
- Answering mathematical questions you have not seen before, with ideas and strategies that you come up with yourself (as opposed to performing calculuations after seeing examples of similar calculations)
- Speaking about mathematics in front of other people
- Communicating careful mathematical arguments using the mathematical typesetting language LaTeX
- Working with with other students to find solutions to problems and to communicate results using the collaborative tool Overleaf
- Getting used to accepting failure as part of the mathematical process, and confidently getting up and trying again after those failures

### Get a preview of future courses

A side benefit of using Discrete Mathematics as the medium for this bridge course is that a number of other mathematical topics have seeds within Discrete Mathematics. During the semester we’ll cover five sections that provide windows into math courses that you may choose to take in the future:

Section |
Course |

3.1 Divisibility | MATH 310: Elementary Theory of Numbers |

4.5 Probability | MATH 318: Probability and Statistics |

5.4 Quotient Spaces | MATH 430: Abstract Algebra |

6.3 Cardinality of Infinite Sets | MATH 315: Real Number System |

7.1 Graphs | MATH 353: Graph Theory |