# 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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essential role in the design of computer circuits. The algebra of sentential logic is based on the notion of logical equivalence and is really quite similar to the standard algebra of numbers and variables. For example, we can expand (2x)2 = 4x 2 using either of the algebraic identities (ab)2 = a2 b2 or (a + b)2 = a2 + 2ab + b2 . Similarly, in the setting of sentential logic, we utilize known logical equivalences to manipulate and simplify formal sentences. In this way, logical equivalence

1.7.3 We prove that for every integer n ∈ Z, if n2 is even, then n is even. An Expanded Proof Taking the contrapositive of “if n2 is even, then n is even” by swapping and negating the premise and conclusion, we obtain “if n is not even, then n2 is not even.” By the parity property of the integers, every integer is either even or odd, and so an integer that is not even must be odd. Therefore, the contrapositive is equivalent (by the parity property) to the implication “if n is odd, then n2 is

properties of limits, derivatives, and Riemann integrals, the deﬁnitions of cardinality and countability, Cantor’s diagonalization arguments to prove the countability of the rationals and the uncountability of the reals, and a brief introduction to L2 spaces. An application section explores how differential equations can model physical processes such as the motion of a clock pendulum. The chapter assumes competency with topics found in a standard single-variable calculus course. As for any of the

properties of limits, derivatives, and Riemann integrals, the deﬁnitions of cardinality and countability, Cantor’s diagonalization arguments to prove the countability of the rationals and the uncountability of the reals, and a brief introduction to L2 spaces. An application section explores how differential equations can model physical processes such as the motion of a clock pendulum. The chapter assumes competency with topics found in a standard single-variable calculus course. As for any of the

D3 is a nonAbelian group, as stated in theorem 2.5.1 below. Working toward a proof that D3 is a group, we ﬁrst consider the closure of D3 under composition—for every a, b ∈ D3 , we want to show that a ◦ b ∈ D3 . Recall from section 2.3 that a Cayley table provides a thorough and systematic approach to computing the action of the binary operation on all possible pairings of elements. If only elements from the given set appear in the corresponding Cayley table, the set is closed under the given