Bridge to Abstract Mathematics: Mathematical Proof and Structures

6.1 CONCLUSIONS INVOLVING V, FOLLOWED BY 3 203 Prove that lirn,,, f(x) exists if and only if both lim,,,+ f(x) and lim,,,- f(x) ...

204 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 6.2 Indirect Proofs An indirect proof is a proof in which we establish the ...

6.2 INDIRECT PROOFS 205 break up the proof into n separate derivations, each of which establishes one of the components q,, that ...

206 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 Note that we do not need to repeat the argument just given with the roles o ...

6.2 INDIRECT PROOFS 207 EXAMPLE 3 Prove that if a linear function y = f (x) = Mx + B is increasing on R, then M > 0. Discussi ...

208 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 The result in Example 5 is of interest in its own right, since it is an imp ...

6.2 INDIRECT PROOFS 209 readily available direct argument, can lead nowhere or to a convoluted argument or at best to an argumen ...

210 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 Two classic proofs by contradiction are the proofs that fi is irrational an ...

6.2 INDIRECT PROOFS 211 (ii) Prove, by the choose method, that if A, B, and C are sets, then A u (B n C) E (A u B) n (A u C). (c ...

212 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 (b) A subset S of R is said to be closed under addition if and only if x + ...

6.3 EXISTENCE AND UNIQUENESS (OPTIONAL) 213 You may recall the notation (3!x)(p(x)) and the accompanying formal definition of "u ...

214 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 in particular, never proves existence of a solution. The reasoning under- l ...

6.3 EXISTENCE AND UNIQUENESS (OPTIONAL) 215 Y, = Y, n U = Y, n (X u Y,) = (Y, n X) u (Y, n Y,) = % u (Yl n Y2) = Y, n Y, so that ...

2l6 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 a point xo in the domain off that is within both 6, and 6, distance of a. F ...

6.3 EXISTENCE AND UNIQUENESS (OPTIONAL) 217 advanced settings we often give a direct proof of existence with reference to some a ...

218 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 One such axiom is the least upper bound axiom for R, stated in Exercise 10. ...

6.3 EXISTENCE AND UNIQUENESS (OPTIONAL) 219 The distinction between the types of existence proof given in Examples 9 and 10 (exi ...

220 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 (c) The additive inverse axiom for R states that, corresponding to every x ...

(b) Let f be a function that maps real numbers to real numbers, and let A and B be subsets of the domain off. Prove that f (A n ...

222 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 (a) Use this theorem, together with the result of Exercise 8, Article 6.2, ...