Bridge to Abstract Mathematics: Mathematical Proof and Structures

10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 343 This exercise relates to the proof of Theorem 7. (a) Prove Theorem 7( ...

344 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 N x N of all ordered pairs of positive integers. Define a relation - on this set b ...

10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 345 answer. Suppose you come along and carry out the same procedure, selec ...

346 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 one (Verify!) and is not onto. In fact, an equivalence class [(p, q)] in Z is in t ...

10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 347 need to have a solution in N. In fact, a solution exists in N if and o ...

348 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 integers except 1 and - 1 also fail to have multiplicative inverses in Z. We next ...

10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 349 a contradiction of Theorem 7(d). If x is negative and xy = 1 for some ...

350 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 issue that must always be attended to whenever we define algebraic op- erations on ...

10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 351 main; this F is called the field of quotients of D. Viewed in this mor ...

352 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 (a) Prove that the relation < from Definition 2 is well defined. That is, prov ...

10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 353 to present are, in all likelihood, slightly beyond the level of experience of ...

354 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 Proof Let E be an arbitrary positive rational number. We must prove that there exi ...

10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 355 that lq,l < B for all n E N. We remark also that once Theorem 2 is known, i ...

358 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 what should be true about the componentwise difference of the two se- quences? Tho ...

10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 357 tempting to fill in the proofs; (b) is by far the less routine and we include ...

358 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 the real number [qb] if and only if (q, - qb) is a null sequence; that is, (q,) is ...

10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 359 equivalence class containing the constant sequence with all terms equal to the ...

360 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 was introduced, in Article 9.3, we promised to provide an example of a non- Archim ...

10.3 OUTLINE OF THE CONSTRUCTION OF THE REALS 361 You need to use both the fact that {a,) is Cauchy and the fact that {a,) is ev ...

Answers and Solutions to Selected Exercises Article 1.1 1.(b) B={-2) (i)={l,2 (m) M={-5,-1,(1&@)/3). (4 R (4 [-0, $1 (4 C ( ...