530 Hwk 1-2 - Math 530 IIWK 1 From Munkres page...

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Unformatted text preview: Math 530 IIWK 1 From Munkres : page l4—l5/1 (for indexed families of sets), 2aefmo,lO page 2O-21/2gh,4ade,53bc Find on; An and Um: An if An: (—3] "7:11 For each (ab) 6 R2, let Ea), be the line with slope a and y-intercept b. Sketch U(a,1))EJEa,b when (a) J: ((a,0) : —I S a S 1}; (b) J = {(2,b) :b S 0}. Let f: R2 —> R be described by f(x,y) = :1. Compute * « (a) f({(x,y) : x2 + 3/2: 1}); (b) f({(x,y) : x = 2y}); (c) f'l-({3}). Let f: @({1,2.3,4}) —> 2 be defined by ft'A) =the number ofelements in A. Thus, f({ l,3,4}) = 3, etc. Compute (a) f({l},{2,3},{2,4}); (b) f']({3}); (C) f_l({'15031})- Given integers a and b, we say that a divides b ifthere exists an integer k such that b = ka. Let f: {1,2,3. . . .,10} —> 2* be defined by f(x) = the number ofelements of 2“ that divide x. Find: (a) f(6); (b) f({6,8}); (c) f‘1({4,5,6, . . .}); (d) f“({2}); (e) the image set of {1,2,3, . . .,10}. Let {Am 0: E J} bea family of subsets ofa set A, and let {qu (X E J} bea family of subsets of a set B, and let 1’: A —> B. Prove : (a) f'l(U0tEJ Bot) = Uale‘1(Ba); (b) f(UaeJ Au) = UaeJ f( Act)- Let f: A —> B and let Bo C B and B] C B. Prove f‘](Bi—B0)=f‘1(BI)—f'l(Bo). Math 530 HWK 2. From Munkres : page 28—29l1,4—6,9 page 39/4-5 page 61—62/2abd,5 page 71/2 I. 011 R2, write (it-1,311) ~ (x2,y2) if yg — y] = 3(‘x2— x1). Show ~ is an equivalence relation, and describe the equivalence classes. (a) Prove that as b if a divides b givesapartial order on 2*. (b) Consider A] = {3,4,5,10,12,60} as a subset of 2+. Does A1 have a largest element? A smallest element? Find all maximal and minimal elements of A], and all upper and lower bounds of A1 with respect to the overall set 2* - (6) Answer the questions of (b) where A1 is replaced by A2 = {4n : n E 2"} ((1) Does division define a partial order on 7. ? ...
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