(BQ) Part 2 book "A transition to advanced mathematics" has contents: Infinite sets, countable sets, the ordering of cardinal numbers, comparability of cardinal numbers and the axiom of choice, algebraic structures, operation preserving maps, completeness of the real numbers, the bounded monotone sequence theorem,.and other contents. | 4/19/10 3:37 PM Page 185 C H A P T E R 4 Functions The notion of a function is familiar to you from previous study in algebra, trigonometry, and calculus. The Preface to the Student reviews the concept of a function as a rule of correspondence and the basic properties of and notations for functions. In this chapter, where we view functions as single-valued relations, our goals are (1) to develop a deeper understanding of methods of constructing functions and the properties of being one-to-one and onto, and (2) to write proofs establishing that a relation is a function, or has (or does not have) these properties. The techniques and results developed here are used throughout the remainder of the text. Functions as Relations The concept of a function is very old, but the word function was not explicitly used until 1694 by G. W. Leibnitz.* It is only relatively recently that it has become standard practice to treat a function as we define it below—as a relation with special properties. This is possible because the rule that makes an element in one set correspond to an element from a second set may be viewed as forming a collection of ordered pairs. * Gottfried Wilhelm Leibnitz (1646–1716) was a versatile German scholar, lawyer, and diplomat who made major contributions to mathematics, philosophy, logic, technology, and physics. Although they worked independently, both he and Isaac Newton developed calculus. Leibnitz devised the now standard dy dx and 1 f ( x )dx notations, referring to dy and dx as “infinitesimals.” His development of the binary number system is the basis of all modern computing devices. 185 Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 186 4/19/10 3:37 PM Page 186 CHAPTER 4 Functions DEFINITIONS A function (or mapping) from A to B is a relation f from A to B such that (i) (ii) the domain of f is A,
