# A Transition to Advanced Mathematics: A Survey Course

## William Johnston

Language: English

Pages: 768

ISBN: 0195310764

Format: PDF / Kindle (mobi) / ePub

A *Transition to Advanced Mathematics: A Survey Course* promotes the goals of a "bridge'' course in mathematics, helping to lead students from courses in the calculus sequence (and other courses where they solve problems that involve mathematical calculations) to theoretical upper-level mathematics courses (where they will have to prove theorems and grapple with mathematical abstractions). The text simultaneously promotes the goals of a ``survey'' course, describing the intriguing questions and insights fundamental to many diverse areas of mathematics, including Logic, Abstract Algebra, Number Theory, Real Analysis, Statistics, Graph Theory, and Complex Analysis.

The main objective is "to bring about a deep change in the mathematical character of students -- how they think and their fundamental perspectives on the world of mathematics." This text promotes three major mathematical traits in a meaningful, transformative way: to develop an ability to communicate with precise language, to use mathematically sound reasoning, and to ask probing questions about mathematics. In short, we hope that working through *A Transition to Advanced Mathematics* encourages students to become mathematicians in the fullest sense of the word.

*A Transition to Advanced Mathematics* has a number of distinctive features that enable this transformational experience. Embedded Questions and Reading Questions illustrate and explain fundamental concepts, allowing students to test their understanding of ideas independent of the exercise sets. The text has extensive, diverse Exercises Sets; with an average of 70 exercises at the end of section, as well as almost 3,000 distinct exercises. In addition, every chapter includes a section that explores an application of the theoretical ideas being studied. We have also interwoven embedded reflections on the history, culture, and philosophy of mathematics throughout the text.

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interpreting the negation connective in a sentence. Instead, the truth table for negation has mathematically formalized the interpretation of negation when computing the truth of sentences. We refer to this truth table when a negation appears in a sentence, an approach which is particularly helpful when working with elaborate compound sentences. By developing similar truth tables for the other logical connectives and capturing our natural intuitions about these connectives, we establish the

translations and developing students’ capability to create natural deductive arguments (rather than just identifying the steps in such arguments as we have done in this text). Copi and Cohen [46], Gustason and Ulrich [105], and Jacquette [127] are widely used texts that support such philosophy courses and enable students to develop their skills in this more computational approach to these logics. There is also a Schaum’s Outline [181] and a more recent abridgment by McAllister [182] that explore

centralizer of a. Consider the following deﬁnition. Deﬁnition 2.5.3 If a is an element of a group G under operation ◦, then the centralizer of a in G is the set of elements in G that commute with a. Symbolically, the centralizer of a in G is denoted by C(a) = {g : g ∈ G and a ◦ g = g ◦ a}. Example 2.5.4 We identify the centralizers of R0 , R120 , and FT in D3 under composition. Since the identity element R0 commutes with every element of D3 , we immediately have C(R0 ) = D3 . In contrast, the

Advanced Mathematics 40 years, the use of formal languages began playing a key role in the design of the computer chips that are so essential to our technologically based society. In this chapter, we develop a formal language in the spirit of Aristotle and Boole known as “sentential” logic. We examine the interaction of sentential logic with our natural language and our intuitive notion of truth. We develop an algebra of sentential logic, explore the expressiveness of this language, consider an

10 inclusive. Every integer between 11 and 20 inclusive. For a ﬁxed, nonprime integer n ∈ Z with prime power factorization n = n pn11 · pn22 · · · pk k , what is the largest possible integer that can appear in this factorization? Explain your answer. Exercises 35–42 consider properties of greatest common divisors. A pair of integers has a greatest common divisor (gcd) (or factor) k when k is the greatest divisor of both. For example, 25 and 40 have a greatest common divisor 5 because 5 is a