These lecture notes are based on the class material “College Discrete Mathematics” for students in the Foundation Semester year at University of Debrecen, Hungary. The lecture notes are intended to help the students understand and learn the course material, but they do not substitute participation and active work on the class.

Discrete mathematics deals with the non-continuous mathematics. This usually means finite mathematics, but properties of natural numbers are discussed, as well. The course sets the basis for future mathematical classes, and is essential to understand those later.

In Chapter 1 we introduce basic mathematical concepts, such as sets, subsets, sums and products, the Euclidean algorithm and numeral systems. In Chapter 2 we show different counting arguments. We count the number of sequences, subsets, permutations, and anagrams. Then we consider the number of ordered subsets, the number of subsets of a given size. Finally, we count the number of possibilities to distribute money, and to take out some balls from an urn.

In Chapter 3 we explain different basic mathematical proof methods, such as mathematical induction and proof by contradiction. We show how one can prove theorems in a constructive way, or by using the pigeonhole principle. At the end of the chapter we use these proof techniques to bring the reader “behind the curtains” of a mathematical card trick.

In Chapter 4 we consider Pascal's famous triangle built up from the binomial coefficients. We prove several identities related to the triangle, and use it to show the Binomial theorem. In Chapter 5 recurrence sequences are discussed. We start by the famous Towers of Hanoi, then work our way up to more general recurrence sequences and to the explicit formulas determining them. Finally, in Chapter 6 we give all solutions to the exercises occurring in the lecture notes.