1549901369-Elements_of_Real_Analysis__Denlinger_

A.4 Properties of Equality 611 Notice that neither of these theorems claims that the statements are all true, only that they are ...

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Appendix B Sets and Functions B.1 Se The conce even prer be able all ob.i logier basi Algebra of Sets .nost basic of all mathema ...

614 Appendix B • Sets and Functions If a set has too many members to list, we can use an ellipsis (three dots), as in {l, 2, 3, ...

B.l Sets and the Algebra of Sets 615 These sets are conveniently illustrated in the following "Venn diagrams:" (a)A u B (b)A n B ...

616 Appendix B • Sets and Functions (g) (Au B)c = N n B C. (de Morgan's law) (h) (An B)c = Ac u BC. (de Morgan's law) (j) uc = 0 ...

B.l Sets and the Algebra of Sets 617 Definition B.1.8 (Operations on Collections of Sets) Let C = {A>. : .A E A} be a collect ...

618 Appendix B • Sets and Functions Part 1: Suppose x EB - ( n A;). Then x EB but x ~ ( n A;). That .AEA .AEA is , x EB but rv \ ...

B.2 Functions 619 (d) {A>.:>. EA}= {[-2 + ~, 2 - ~) : n EN}. ( e) {A>. : >. E A} = { ( n, n + 1) : n E N}. Prove Th ...

620 Appendix B 11 Sets and Functions Definition B.2.2 Two functions f : A ~ B and g : A' ~ B' are said to be equal if A= A', B = ...

B.2 Functions 621 The set f(C) is called the image of C under f and the set f-^1 (D) is called the inverse image of D under f. W ...

622 Appendix B • Sets and Functions (e) VDi, Dz<:;; B, f-i(Di n Dz)= f-i(Di) n f-i(Dz). (f) VDi, Dz<:;; B , ri(Di - Dz)= 1 ...

y=x^2 +3 -1 1 2 x Figure B.5 (e) f-^1 ((-oo,3] n (2,4)) = f-^1 (2,3] = {O}, and r^1 (-oo, 3] n f-^1 (2, 4) = {O} n (-1, 1) = {O} ...

624 Appendix B 11 Sets and Functions GRAPHS OF FUNCTIONS f : JR--+ JR Definition B.2.14 The graph of a function f : A __, B , wh ...

Let l(x) = 4 - x^2. Find (a) 1J(f) (c) l[O, 1] (e) 1(0, 2) (g) 1-^1 [2, 4] (i) 1-^1 (0, 00) (k) 1-^1 ({0}) Let l(x) = 2x. Find ...

626 Appendix B • Sets and Functions We define algebraic operations on this set :F(S, JR). In particular, we define (a) Addition: ...

B.3 Algebra of Real-Valued Functions 627 (g) Minimum: V f, g E F(S, JR), we define the function min{!, g} by specifying that Vx ...

628 Appendix B • Sets and Functions (11) VJ,g E F(S, JR), Jg E F(S, JR); (12) VJ, g, h E F(S, JR), f(gh) = (fg)h; (13) VJ, g E F ...

B.3 Algebra of Real-Valued Functions 629 (12) Let f , g, h E F(S, JR). Then, \:/x ES, [f(gh)](x) = f(x) · (gh)(x) by definition ...

630 Appendix B 11 Sets and Functions BEWARE! Notice the reversal of orientation: In the schematic drawing, the function f is dra ...