Bridge to Abstract Mathematics: Mathematical Proof and Structures

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 363 6.(a)(j)AnC=[2,4]=C-A', (f)A-C=(4,9), (g) C - A= [-l,2) (c) A- B=(-a, -31 (d) A ...

364 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES (true), q: September is the name of a month (true), r: Wednesday is the name of ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 365 Article 3.2 (d) All easy problems are solvable. (i) Some unsolvable problems ar ...

366 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES Hx): X is a field, q(x): X is a ring, r(x): X is an integral domain; this argum ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES^367 (b) Proof Assume C is a set satisfying C E A, for each k = 1,2,3,.... To prove ...

368 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES (d) Proof Let A be an arbitrary subset of U. Then, A A U = (A - U) u (U - A) = ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 369 Article 5.3 (a) Proof Let X, A, and B be sets such that X c B. To prove X u (A n ...

370 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 3m2 + 3m + 1 < 3m2 + 3mZ + 3m2 = 9m2 < m(m2) = m3. Hence m3 + 3m2 + 3m + 1 ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES^371 8. 9. IS. x E A - B. To do this, suppose x were an element of B. Then we ...

372 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES positive integer M, such that Ix, - x) < E whenever n 2 M. Hence, for the des ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 373 R = R-l. To prove R is symmetric, let (x, y) E R. Then (y, x) E R-' = R, so (y, ...

374 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES (c) Proof Assume [x] = [y]. Then y E [x], so that x E y. Conversely, suppose x ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 375 (-n/2, d2). The latter is one to one over that interval and has range R. Hence t ...

376 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES (a) Proof Suppose 0 is not finite, that is, 0 is infinite. Then there exists a ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES 377 y = x, then clearly ax = ay. So assume y - x E 8 and -a E 8. Then the product (y ...

378 ANSWERS AND SOLUTIONS TO SELECTED EXERCISES Article 10.1 (b) x1 =x; xn= x(9-') ifn > 1. (b) Proof Let S = {m~Nlo(m) = 1 ...

ANSWERS AND SOLUTIONS TO SELECTED EXERCISES^379 have b < a or a > b, as desired. The converse follows by reversing each of ...

List of Symbols Symbol Meaning Set membership and nonmembership, p. 4 Roster description of a finite set, p. 4 Rule description ...

LISTS OF SYMBOLS 381 Logical negation, conjunction, disjunction connectives, p. 56 Conditional, biconditional connectives, p. 61 ...

382 LISTS OF SYMBOLS Arbitrary union and intersection (resp.), p. 290 Direct product of an arbitrary collection, p. 291 A field ...