Read e-book online An Introduction to Abstract Mathematics PDF

By Robert J. Bond

ISBN-10: 1577665392

ISBN-13: 9781577665397

Bond and Keane explicate the weather of logical, mathematical argument to clarify the that means and significance of mathematical rigor. With definitions of thoughts at their disposal, scholars study the principles of logical inference, learn and comprehend proofs of theorems, and write their very own proofs--all whereas turning into acquainted with the grammar of arithmetic and its variety. furthermore, they're going to strengthen an appreciation of the several equipment of evidence (contradiction, induction), the price of an evidence, and the great thing about a chic argument. The authors emphasize that arithmetic is an ongoing, bright discipline--its lengthy, interesting heritage regularly intersects with territory nonetheless uncharted and questions nonetheless wanting solutions. The authors' broad historical past in instructing arithmetic shines via during this balanced, specific, and fascinating textual content, designed as a primer for higher-level arithmetic classes. They elegantly exhibit technique and alertness and realize the byproducts of either the achievements and the missteps of prior thinkers. Chapters 1-5 introduce the basics of summary arithmetic and chapters 6-8 practice the tips and strategies, putting the sooner fabric in a true context. Readers' curiosity is constantly piqued via transparent reasons, functional examples, dialogue and discovery workouts, and old reviews.

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EXAM P L E 5 P: Q: Gerald Ford was vice president under Jimmy Carter. 2 < 7. Here, P is false and Q is true, so P � Q is true. Notice that if R is the statement 7 < 2, P � R is also true; a false statement implies anything ! EXAM P L E 6 P: Q: The American Revolution ended in 178 1 . George Washington served three terms as president. P is true and Q is false; however related the statements might be, the statement P � Q is false. 3 P: Q: EXAM PLE 7 31 I M PLI CATI O N S 4 is an even integer. 42 is an even integer.

Use inductive reasoning to find a statement about whether or not the sum of two consecutive even integers is divisible by 4. Then prove your statement. D4. Criticize the following "statement" of the Fundamental Theorem of Calculus: f(x) dx = F(b) - F(a). J: D5. Criticize the following "proof" of the fact that if n and m are even then n + m is even. We know that n = 2t and m Therefore, n + m is even. = 2t, so n + m = 2t + 2t = 4t. Write out a correct proof. D6. Is the following a valid proof that every integer multiple of 4 is even?

We leave it as an exercise to show that these statements have the same truth tables. " 34 CHAPTER I MATH EMATICAL REAS O N I N G The value assigned to the variable x that makes P(x) true and Q(x) false is called a counterexample to the statement "For all x, P(x) => Q(x) . " Note that the negation of a n implication is not a n implication! EXAM P L E 1 3 Consider the statement "The sum of two perfect squares is a perfect square. " T o prove o r disprove this statement, let's first write i t a s a statement that has a universal quantifier and contains an implication: "For every pair of integers n and m, if n and m are perfect squares, then n + m is a perfect square.

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An Introduction to Abstract Mathematics by Robert J. Bond

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