Bridge to Abstract Mathematics: Mathematical Proof and Structures

8.3 CARDINAL NUMBER OF A SET 283 element x in (0, 1) can be represented by a decimal expansion, say, x = d,d,d,.... We can guara ...

284 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 interval on the real line (see Exercise 1) are uncountable, and in fact have the ...

8.3 CARDINAL NUMBER OF A SET 285 either x E f(x) or x 4 f(x) for each x E S. Consider the set N = (x E S 1 x 4 f (x)}. Clearly N ...

286 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 (although if you worked through Exercise 18, Article 8.2, you should find it fam ...

8.3 CARDINAL NUMBER OF A SET 287 Proof Since (0, 1) z R, by Exercise 1, we may just as well prove that (0, 1) 9(N). We will show ...

288 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 Results about countably infinite sets: (a) If n E N, then n < KO. (b) If A ...

8.4 ARBITRARY COLLECTIONS OF SETS 289 (a) Prove that the set Z of all integers is countably infinite (recall Figure 8.7). (b) P ...

290 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 contain only a countable number of sets. Thus the collections of sets con- sider ...

8.4 ARBITRARY COLLECTIONS OF SETS 291 EXAMPLE 1 Let f: X -, Y. Prove that the image under f of the intersec- tion of any collect ...

292 RELATIONS: FUNCTIONS AND MAPPINGS Chapter 8 Suppose at = (A$ E I) is a collection of sets and J c I. Prove: Suppose at = (A ...

Properties of the Number Systems of Undergraduate Mathematics CHAPTER 9 Throughout most precalculus and elementary calculus cour ...

294 PROPERTIES OF NUMBER SYSTEMS Chapter 9 "Write down what you know about the rational number system,... the in- tegers,... the ...

9.1 FIELDS 295 multiplication of n x n matrices, and composition of mappings from a given set X into itself. An operation may be ...

298 PROPERTIES OF NUMBER SYSTEMS Chapter 9 10. I II. the field, satisfying 1. a = a for all a E F To each a E F, there correspon ...

9.1 FIELDS 297 EXAMPLE 4 For those familiar with a little matrix theory, the set of all 2 x 2 matrices, with operations of ordin ...

298 PROPERTIES OF NUMBER SYSTEMS Chapter 9 Figure 9.2 (a) Tables defining the field of integers modulo 3; (b) tables defining th ...

9.1 FIELDS 299 of R is analogous to that of our other examples of fields now; in particular, we must verify (i.e., prove) the 11 ...

300 PROPERTIES OF NUMBER SYSTEMS Chapter 9 Proof By the uniqueness of the additive inverse -x of a field element x (Theorem I), ...

9.1 FIELDS 301 Analogs to the familiar subtraction and division operations from R exist in an arbitrary field. DEFINITION 3 Let ...

302 PROPERTIES OF NUMBER SYSTEMS Chapter 9 w-(x-y)= 1-6= 1 +(-6)= 1 + 1 =2. Ontheotherhand,w-x= w+(-x)= 1 +(-5)= 1+2=3,so(w-x)+y ...