Bridge to Abstract Mathematics: Mathematical Proof and Structures

5.4 PROOF BY MATHEMATICAL INDUCTION 1m Generalization. Many theorems or axioms whose familiar statement says something about two ...

184 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 The resblts in Examples 5 and 6, when combined with an elementary summation ...

5.4 PROOF BY MATHEMATICAL INDUCTION 185 DEFINITION 1 A subset S of the set N of all positive integers is said to be inductive if ...

186 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 least one positive integer no, then it is true for infinitely many positive ...

5.4 PROOF .BY MATHEMATICAL INDUCTION 187 (a) Prove, by induction, that 3 divides 4" - 1 for all n E N. (b) Prove that, for any ...

188 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 let {A,, A,,... , A,,,) be a collection of m + 1 sets. We claim that A, = A, ...

5.4 PROOF BY MATHEMATICAL INDUCTION 189 Finally, we have [sin 2mx + 2 sin x(cos 2mx cos x -'sin 2mx sin x)]/2 sin x = [sin 2mx - ...

Methods of Mathematical Proof, Part 11. ~dvanc& Methods CHAPTER 6 The question of what constitutes an advanced, as opposed t ...

6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 191 in Article 4.3. But in junior-senior level mathematics, definitions involving 3 a ...

192 METHODS OF MATHEMATICAL PROOF, PART ll Chapter 6 EXAMPLE 6 Let q(x, y), r(x, y, z), and s(x, y, z) be propositional function ...

6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 193 chosen element of U and assuming, that r(x, y, z) is true. The final step is to c ...

194 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 Solution Let A and B be given invertible n x n matrices. We must find a mat ...

6.1 CONCLUSIONS INVOLVING V FOLLOWED BY 3 195 S n T. The key is the choice of 6; we can go no further in the proof until we've d ...

196 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 We deal next with union. To prove S u T is open, let x E S u T. To prove th ...

6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 197 EXAMPLE 10 Prove that lim,,, (4x + 7) = 19. Solution Note that f(x) = 4x + 7, a = ...

198 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 In most theorems about general properties of limits the hypotheses in- clud ...

6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 199 it works! The best answer we can give to the second question is that most student ...

200 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 (b) An integer n is said to be even if and only if there exists an integer ...

6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 2M (a) Prove that 3 is an interior point of [O, 1). (b) Prove that 1 is not an inter ...

202 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 Use an epsilon-delta argument to prove: (a) lim,,,(-3x + 6) = -6 (b) lim,, ...