By Douglas Smith, Maurice Eggen, Richard St. Andre
A TRANSITION TO complicated arithmetic is helping scholars make the transition from calculus to extra proofs-oriented mathematical research. the main winning textual content of its variety, the seventh variation maintains to supply a company origin in significant recommendations wanted for endured research and courses scholars to imagine and convey themselves mathematically--to learn a scenario, extract pertinent evidence, and draw acceptable conclusions. The authors position non-stop emphasis all through on bettering students' skill to learn and write proofs, and on constructing their severe expertise for recognizing universal mistakes in proofs. thoughts are basically defined and supported with specific examples, whereas plentiful and various workouts supply thorough perform on either regimen and tougher difficulties. scholars will come away with a superb instinct for the kinds of mathematical reasoning they'll have to practice in later classes and a greater figuring out of the way mathematicians of every kind technique and clear up difficulties.
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Extra resources for A Transition to Advanced Mathematics (7th Edition)
Since both a and b are positive, b + a > 0. Since a < b, b − a > 0. Because the product of two positive real numbers is positive, (b − a)(b + a) > 0. Therefore b2 − a2 > 0. Ⅲ It is often helpful to work both ways—backward from what is to be proved and forward from the hypothesis—until you reach a common statement from each direction. Example. Prove that if x 2 ≤ 1, then x2 − 7x > − 10. Working backward from x2 − 7x > −10, we note that this can be deduced from 2 x − 7x + 10 > 0. This can be deduced from (x − 5)(x − 2) > 0, which could be concluded if we knew that x − 5 and x − 2 were both positive or both negative.
We begin with an initial set of statements, called axioms (or postulates), that are assumed to be true. We then derive theorems that are true in any situation where the Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. qxd 28 CHAPTER 1 4/22/10 1:42 AM Page 28 Logic and Proofs axioms are true. The Pythagorean* Theorem, for example, is a theorem whose proof is ultimately based on the five axioms of Euclidean† geometry. In a situation where the Euclidean axioms are not all true (which can happen), the Pythagorean Theorem may not be true.
4 Basic Proof Methods I 31 The previous example shows the power of pure reasoning: It is the forms of the propositions and not their meanings that allowed us to make the deductions. Because our proofs are always about mathematical phenomena, we also need to understand the subject matter of the proof—the concepts involved and how they are related. Therefore, when you develop a strategy to construct a proof, keep in mind both the logical form of the theorem’s statement and the mathematical concepts involved.
A Transition to Advanced Mathematics (7th Edition) by Douglas Smith, Maurice Eggen, Richard St. Andre