04101 Introduction to Mathematics
Credits: 6 introductory credits in Mathematics
Authors: Daniela Leibowitz, Amos Altshuler, Ehud Artzi, Maxim Bruckheimer, Yair Tauman, Itzhak Zikoni
The objectives of this course are to provide beginners in mathematics with some fundamental mathematical tools, to acquaint them with abstractions and mathematical ways of reasoning, and to develop their ability to rigorously formulate and prove mathematical statements. The course attempts to achieve these goals through discussions presented in reasonable depth, without assuming prior proficiency in mathematics. It does not require of its students, nor does it develop in them, the high school level mathematical skills required for most of the other mathematics courses taught at the University. However, it is expected to facilitate the study of further university-level Mathematics courses.
Topics: Sets sets and their elements; finite and infinite sets; operations on sets (unions, intersections and complements); relations between sets (inclusion, equivalence); non equivalent infinite sets; Cardinal numbers; Binary operations basic definitions; associativity; commutativity; neutral elements; inverse elements; groups; Functions introduction; Cartesian products; functions as subsets of Cartesian products; graphs; composition of functions; injective, bijective and surjective functions; invertible functions; arithmetic operations on real functions; groups of functions; Plane-isometries definition; reflections, translations and rotations; compositions of reflections; congruent triangles and isometry; classification of the plane isometries; the group of plane isometries; applications in Euclidean geometry; Axiom systems undefined terms, axioms and theorems of axiomatic mathematical theories; models; consistency, independence, completeness and categoricity of axiom systems; finite affine geometries; Euclidean geometry historical notes, a modern (partial) representation of Euclidean plane geometry as an axiomatic theory; The natural numbers an introduction to Peanos axioms, arithmetic operations, order relations; first steps in number theory; mathematical induction.