Number Theory

Reading, Writing, and Proving: A Closer Look at Mathematics by Ulrich Daepp

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By Ulrich Daepp

The reader of this publication is perhaps approximately to coach or take a "first path in evidence techniques." scholars are taking this direction simply because they prefer arithmetic, and the authors desire to maintain it that method. At this element, they've got an intuitive feel of why issues are real, yet no longer the publicity to exact and demanding considering essential to live on within the mathematical international. The authors have written this e-book to bridge the distance. usually, scholars starting this path have little education in rigorous mathematical reasoning; they want counsel. on the finish, they're the place they need to be; on their lonesome. The authors target is to educate the scholars to learn, write and do arithmetic independently, and to do it with readability, precision, and care.

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A) For every , there exists δ such that δ < . (b) Let a ∈ R. For every there exists δ such that for every x ∈ R, if |x − a| < δ, then |x2 − a2 | < . (c) Let x ∈ R. Then x < x + for all > 0. 52 4. Set Notation and Quantifiers (d) For every integer n, there exists x > n such that x2 > n2 . (e) For every > 0 there exists an integer N such that 1/n < for all n ≥ N. (f) For all x, either x < 0 or x > 0. (g) For all x, there exists an integer n such that n > x. (h) For all x, y, and z, if x < y and z < 0, then zx > zy.

Let x be even. Then there is an integer n such that x 2n. Therefore, x2 (2n)2 2(2n2 ). Let m 2n2 . Then x2 2m and m is an integer. Therefore x2 is even. 34 3. Introducing the Contrapositive and Converse Of course the original theorem is now also proven since it is equivalent to the one we proved. Thus, using the contrapositive is one possible way to attempt to prove that an implication is true. We will soon have a number of ways to attack a problem. Try to keep them all in mind. Some other related remarks: Notation is more important than it may seem.

Introducing the Contrapositive and Converse Of course the original theorem is now also proven since it is equivalent to the one we proved. Thus, using the contrapositive is one possible way to attempt to prove that an implication is true. We will soon have a number of ways to attack a problem. Try to keep them all in mind. Some other related remarks: Notation is more important than it may seem. In the theorem above, we assume that x is even and try to show x2 is even. If we assume that x 2n and accidentally 2 2 try to show x 2n (rather than x 2m), we’re stuck because we erroneously assumed that x x2 .

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