Bridge to Abstract Mathematics: Mathematical Proof and Structures

4.3 THE LIMIT CONCEPT (OPTIONAL) 143 sin x, x rational (d) = { 0, x irrational I ata=O sin x, x rational = { 0, x irrational at ...

144 ELEMENTARY APPLICATIONS OF LOGIC Chapter 4 *(a) Give an epsilondelta argument that &,,+ sin (l/x) # 0. (b) How might an ...

4.3 THE LIMIT CONCEPT (OPTIONAL) 145 depends on E; specifically, the smaller E is chosen, the larger will be corresponding value ...

Methods of Mathematical Proof, Part I: Elementary Methods CHAPTER 5 As undergraduate students of mathematics pass through the so ...

METHODS OF MATHEMATICAL PROOF, PART 1 147 But a case can be made that students should not view proof writing to be so difficult ...

148 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 to, or can quickly be grasped by, students at the sophomore level of under- ...

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR +. 149 Figure 5.1 Triangles ABO and DCO, jkom Example I, can be proved to be congruent ...

150 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 that two quantities, say, A and 2, are equal, we have provided a string of e ...

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR 4. 151 cluttered and boring. There is no universally correct answer; what con- stitute ...

152 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 by transitivity. The string should begin with (1 + sin x)/cot2 x, end with s ...

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR +. 153 Solution Before attempting a general proof of an unfamiliar theorem, you should ...

154 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 1 solution Let x,y, and z be given. Then Ix - zI = (x + (-y + y) - zl = I(x ...

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR + 155 be possible, if not practical, to prove the theorem by enumerating all the cases ...

156 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 Throughout Exercises 3 through 6, let all sets involved be subsets of a univ ...

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR -, 157 it can be proved that for any real number x and positive real number a, 1x1 r a ...

158 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 Let U = (1,2,3). Consider the assertion that for all subsets A, B, and C of ...

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR -, 159 (f) Prove that if A is any square matrix, then A + At is symmetric, whereas A - ...

180 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 (b) THEOREM For any sets A and B, A' - B' = B - A. "Proof" Let A and B be ar ...

5.2 CONCLUSIONS INVOLVING V AND -+, BUT NOT 3 161 EXAMPLE 5 (Some linear algebra background is helpful.) A set of vectors {v,, v ...

162 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5 EXAMPLE 7 Prove that if M > 0, then the linear function y = f(x) = Mx + B ...