Theorems and Definitions in Mathematics
Teodor Dumitru Vălcan
Abstract
Teaching - learning Mathematics in school assume mastering and the use of mathematical language. It is made up of the natural language of the spoken language (of teachers and students) plus a series of mathematical signs and symbols, but also a series of specific propositions. In this paper we will present two fundamental concepts of mathematical language; namely, the theorem and the definition respectively. Thus, we will present here, all types of mathematical propositions (direct, reciprocal, contrary and reciprocal of the contrary or contrary of the reciprocal), which if true become theorems (direct, reciprocal, contrary and reciprocal of the contrary or contrary of the reciprocal). Regarding the concept of definition, we will refer to its two component parts (the proximate genus and the specific difference) and to the two types of definitions (natural and / or artificial). Finally, we will analyze and exemplify the concept of theorem of characterization. We will exemplify all this with propositions / theorems from Algebra, Mathematical Analysis and Geometry.