Titlebook: Elements of Mathematics; A Problem-Centered A Gabor Toth Textbook 2021 The Editor(s) (if applicable) and The Author(s), under exclusive lic

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Preliminaries: Sets, Relations, Maps,In this chapter we give an account on the foundations of mathematics: naïve and axiomatic set theory. We introduce here several concepts that will play principal roles later: The Least Upper Bound Property for ordered sets, relations, maps, infinite sequences, the principle of inclusion-exclusion, cardinality, and classes vs. sets.

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https://doi.org/10.1007/978-3-642-34795-5 are introduced using Peano’s system of axioms. Inherent in the last Peano axiom is his Principle of Induction, one of the fundamental postulates of arithmetic on natural numbers. Among the myriad of applications of this principle, we discuss here the Division Algorithm for Integers along with the greatest common divisor and prime factorization.

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Real Numbers,leads naturally to Dedekind’s original proof of irrationality of the square root of a non-square natural number. As an immediate byproduct, this implies that the Least Upper Bound Property fails. Another advantage of this proof is that it leads directly to the concept of Dedekind cuts, and thereby t

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Rational and Real Exponentiation, arithmetic properties of the limit inferior and limit superior and (thereby) the limit. The Fibonacci sequence, the geometric and .-series, and some of their contest level offsprings serve here as illustrations. The core material of this chapter proves the existence of roots of (positive) real numb

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Real Analytic Plane Geometry,s path; and, in making use of the real number system already in place, we develop real analytic plane geometry using Birkhoff’s axioms of metric geometry. One of the main purposes of this chapter is to explain what is classically known as the Cantor–Dedekind Axiom: The real number system is order-is

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Polynomial Expressions,It is presented here with full arithmetic and historical details, with many identities, and along with its principal, mostly combinatorial, applications including Bernoulli’s derangements. The Division Algorithm for Integers discussed in Section . leads directly to its polynomial analogue, the Divis