txt Foundations of Mathematics (English Edition)

This book is based on lectures developed by the author to B.Sc and M.Sc. students at Koya University, Department of Mathematics. In addition, the book is also the product of the observations accumulated in the last two decades of teaching under, and graduate studies of the author. The book takes into consideration, the necessity of the contents of this book for students to study mathematics as well as physics in both theoretical and practical branches in the faculties of science, education, engineering, and the statistics department in the faculties of administration and economics. In addition, even faculties of medicine, technical science, industrial mathematics, petrochemical departments, ...etc for their purposes to use the preliminaries of mathematics of some special functions as a tool to implement some tasks in the eld of their professional applications in the eld of mathematics practical work, such as; some functions related to transformations, statistical and probabilistic mappings, and what is related to applied eld. The academic goals of this book are;
(i) To help students to be fully familiar with the foundations of mathematics.
(ii) To help students to use mathematics logically in life to scientific thinking in order to solve problems.
(iii) To employ mathematics in other sciences to facilitate their tasks.
(iv) To help students to study problems from different scientific perspectives. And to find appropriate scenarios to state the algorithms for optimal solutions through logical reasoning and the rule of conditional proof associated with the deductive rules for those problems.
It is noteworthy that, most of the theorems, corollaries, and exercises in this book are adapted from the references (Albert, 1956; Bittinger, 1970; Bittinger, 1985; Birkho and Mac, 1962; Birkho and Mac, 2017; Cohen, 2008; Cohen and Ehrlich, 1969; Eves and Newsom, 1958; Fraenkel, 1969; Hafstrom, 2013; Hall, 2018; Halmos, 2017a; Halmos, 2017b; Herstein, 2006; Hu, 1965; Kamke, 1950; Kelley, 2017; Kelley, 1955; Monk, 1973a; Monk, 1973b; Pervin, 1964; Pervin, 2014; Pinter, 2014; Stoll, 1960; Stoll, 1979; Van der Waerden et al., 1950; Suppes, 1999; Wilder et al., 2012; Wilder, 1952; Zariski and Samuel, 1958; Zariski and Samuel, 2013; Zulauf, 1969b; Zulauf, 1969a; Nagornyi, 1971).
The contents of this book are organized as follows: chapter 1, dedicated to discussing to the mathematical logic and the basic concepts of it. Chapter 2, deals with the sets and operations on them. Chapter 3, deals with relations on sets. The mappings or functions from a set to another set based on the domain and the codomain took their place in chapter 4. Chapter 5, dealing with the potency of sets, equipotent sets, arithmetic on cardinal numbers, ordinal numbers, and paradoxes. Chapter 6, deals with the natural numbers, Peanos axioms, arithmetic of the natural numbers, and innite sets. Chapter 7, considered binary operations and groups, subgroups, Lagrange theorem on groups, and homomorphism and isomorphism. Chapter 8, deals with integers, construction of integers, integers with two binary operations to creation integral domain, rings, and order on integers. Chapter 9, describes how to create rational numbers of integers. Moreover, proves that this collection of numbers with the addition and multiplication operations can be an incomplete Archimedean eld. Chapter 10, collection the rational numbers with irrational numbers to create the real numbers. Finally, chapter 11, expands the real numbers to obtain the eld of the complex numbers, and proves that this kind of the set is closed algebraically.
Theorems and their corollaries are printed in italics. While, the end of the proofs to theorems and corollaries are indicated by the symbol ◆.