Some specific concepts and tools of Discrete Mathematics

Discrete Mathematics deals with finite or countable sets, and thus, in particular, with natural numbers. This mathematical field bring into play several overlapping domains, e.g. number theory (arithmetic and combinatorics), graph theory, and combinatorial geometry. As a consequence of the peculiarities of discreteness versus continuum, interesting specific reasonings can be developed (Batanero and co 1997), and new tools can be constructed, such as coloring, proof by exhaustion of cases, proof by induction, use of the Pigeonhole principle (Grenier et Payan 1999 and 2001). Furthermore, several concepts involved in other mathematical domains are also used in this field, in a particular manner, e.g. the Bijection Principle, optimization techniques and the notions of « generating set » or « minimal set ». In this paper, I wish to develop two of these specific tools, the Pigeonhole Principle and the Finite Induction Principle. I frequently use these tools in my courses, to introduce students to discrete mathematics and initiate counting, along with modelling and proof-elaboration activities.