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 learn. the main profitable textual content of its sort, the seventh variation maintains to supply a company starting place in significant recommendations wanted for endured research and publications scholars to imagine and show themselves mathematically--to study a state of affairs, 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 understanding for recognizing universal error in proofs. techniques are truly defined and supported with specific examples, whereas ample and various routines supply thorough perform on either regimen and more difficult difficulties. scholars will come away with an effective 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 all types method and resolve difficulties.
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Extra info for A Transition to Advanced Mathematics (7th Edition)
2 1. Identify the antecedent and the consequent for each of the following conditional sentences. Assume that a, b, and f represent some fixed sequence, integer, or function, respectively. ૺ (a) If squares have three sides, then triangles have four sides. (b) If the moon is made of cheese, then 8 is an irrational number. (c) b divides 3 only if b divides 9. ૺ (d) The differentiability of f is sufficient for f to be continuous. (e) A sequence a is bounded whenever a is convergent. ૺ (f) A function f is bounded if f is integrable.
Real numbers) 2. For each of the propositions in Exercise 1, write a useful denial, and give a translation into ordinary English. 3. Translate these definitions from the Preface to the Student into quantified sentences. (a) The integer x is even. (b) The integer x is odd. Copyright 2011 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. qxd 26 CHAPTER 1 4/22/10 1:42 AM Page 26 Logic and Proofs 4. ଁ 5. 6. ૺ (c) The integer a divides the integer b.
Which of the following are true? ૺ (a) Triangles have three sides iff squares have four sides. (b) 7 + 5 = 12 iff 1 + 1 = 2. ૺ (c) b is even iff b + 1 is odd. ) (d) m is odd iff m2 is odd. ) (e) 5 + 6 = 6 + 5 iff 7 + 1 = 10. (f) A parallelogram has three sides iff 27 is prime. (g) The Eiffel Tower is in Paris if and only if the chemical symbol for helium is√H. √ √ √ √ √ √ √ 10 + 13 < 11 + 12 iff 13 − 12 < 11 − 10. (h) (i) x 2 ≥ 0 iff x ≥ 0. ) (j) x 2 − y 2 = 0 iff (x − y)(x + y) = 0. ) (k) x 2 + y 2 = 50 iff (x + y) 2 = 50.
A Transition to Advanced Mathematics (7th Edition) by Douglas Smith, Maurice Eggen, Richard St. Andre