Bridge to Abstract Mathematics: Mathematical Proof and Structures

6.4 PREVIEW OF ADDITIONAL ADVANCED METHODS OF PROOF (OPTIONAL) 223 of the mathematician's craft. In all likelihood, if you are t ...

224 METHODS OF MATHEMATICAL PROOF, PART II Chapter 6 interval is bounded on that interval" (which has as a corollary the result ...

BOOK - - - - TWO Bri dging Topics: Relations, Functions. and 1 Number System ...

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Relations, Part Ia Equivalence elations and Partial Orderings CHAPTER 7 In the next two chapters we study three related, yet div ...

228 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 of fractions such as $ and & as the same for purposes o ...

7.1 RELATIONS 229 DEFINITION 1 (a) We say that two ordered pairs of objects (a, b) and (y, z) are equal, denoted (a, b) = (y, z) ...

230 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 (c) If A x B = B x A, A # 0, B # a, then A = B (d) If A x B ...

7.1 RELATIONS 231 EXAMPLE 2 Define a relation R5 on R by R5 = {(x, y) E R x R 13 < x < 8, -2 I y < 4). This relation ma ...

232 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 ordered pair contained in the relation. Using this approach ...

7.1 RELATIONS 233 numbers x, y, and z. The relation is antisymmetric, although only by means of a logical technicality. The ques ...

234 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 Partial proof We consider the "only if" part of (e). Suppos ...

7.2 EQUIVALENCE RELATIONS 235 Find the domain and range of each of the following relations: (a) R, and R,, Example 1 (b) Rs, Ex ...

236 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 (b) Recall the relation R,, from Example 6, Article 7.1. Tw ...

7.2 EQUIVALENCE RELATIONS 237 Solution To show E is reflexive, let x E A be given. Note that (x, x) E E means that x2 > 0, cl ...

238 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 Solution Let A= (1,2,3) and define R,, on A by R,, = ((1, 1 ...

7.2 EQUIVALENCE RELATIONS 239 erties. The ideas in Example 5 are the basis for both Exercise 10 and for our work in the next art ...

240 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 (6) Let F be the set of all real-valued functions having do ...

7.3 EQUIVALENCE CLASSES AND PARTITIONS 241 Condition (i) states that each cell is nonempty. Condition (ii) asserts that any two ...

242 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7 We have just seen that any partition of a set A leads autom ...