An Introduction to Abstract Mathematics by Robert J. Bond

By Robert J. Bond

Bond and Keane explicate the weather of logical, mathematical argument to clarify the that means and significance of mathematical rigor. With definitions of ideas at their disposal, scholars examine the foundations of logical inference, learn and comprehend proofs of theorems, and write their very own proofs--all whereas turning into conversant in the grammar of arithmetic and its kind. furthermore, they'll improve an appreciation of the several tools of evidence (contradiction, induction), the price of an explanation, and the wonderful thing about a chic argument. The authors emphasize that arithmetic is an ongoing, shiny discipline--its lengthy, interesting heritage regularly intersects with territory nonetheless uncharted and questions nonetheless wanting solutions. The authors' wide heritage in educating arithmetic shines via during this balanced, specific, and interesting textual content, designed as a primer for higher-level arithmetic classes. They elegantly exhibit procedure 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 observe the information and strategies, putting the sooner fabric in a true context. Readers' curiosity is consistently piqued by way of transparent factors, useful examples, dialogue and discovery workouts, and old reviews.

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Then . . " using symbols to represent variables. ( b ) Write the negation of the statement, again using symbols. ( c ) Prove the statement if you think it is true or disprove it if you think it is false. 5. Repeat Exercise 4 for the statement "The cube of an odd integer is odd. " 6. Repeat Exercise 4 for the statement "The sum of the squares of two consecutive integers is odd. " 7. Repeat Exercise 4 for the statement "The sum of the squares of three consecutive integers is even. " 8. Repeat Exercise 4 for the statement: "If the square root of a positive integer is an even integer, then the integer itself is even.

But Ptolemy assumed the following statement: Given a point P in the plane and a line L that does not intersect P, there is one and only one line that passes through P and is parallel to L. This statement is actually equivalent to Euclid's fifth postulate and is the version of the fifth postulate that actually appears in most high school geometry texts. The following statements, familiar to students of plane geometry, are also equivalent to the Parallel Postulate: 1. There exist two triangles that are similar but not congruent.

1\ Q) 1\ (P v R) 1\ (P V S ) are v (P 1\ R) v (P 1\ S ) are 16. Let P and Q be statement forms. ( a ) Prove that (P 1\ Q) v (P 1\ -,Q) V (-,P 1\ Q) v (-,P 1\ -,Q) is a tautology. ( b ) Prove that (P v Q) 1\ (P v -,Q) 1\ (-,P v Q) 1\ (-,P V -,Q) is a contradiction. D i s c u s s i o n a n d D i s c ov e ry E x e r c i s e s Dl. Consider the following "proof" that i f n o r m is a n odd integer, then nm is an even integer. Suppose that n is odd and m is even. Then m 2t for some integer t. Therefore nm = n(2t) = 2nt, which is even.

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