2017-08-16 23-57.pdf - UH Math 4377 Dr Heier Spring 2010 HW...

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Unformatted text preview: UH - Math 4377 - Dr. Heier - Spring 2010 HW 1 —— due 01/28 at the beginning of class 1. Let A z: {1, 2, 3}, B = {3, 4}. Write down all elements of the sets AUB, AnB, A\B, Ax A, Ax B. U?“ ‘6 ANKSRUQ \35 ‘t 2. Let z,y E Z. Let 3 ~ y if and only if4Iy—az. Prove that N is an equivalence relation. JNIS, “c kg 1+ Knowing numbers we WSW“?— 3. Let f I {1,2,3,4} *N 7% HM" (a) Find the domain, codomain and range of f. 7(b) ”ES f one—to—one?(‘m3 ethic) (c) Isff ontoflSquecfivQ 1;“ ‘ I 4. Let a, b be arbitrary elements in a field. Prove that («—a.) - b = ~(a. - b). (Hint: You may use without 'proof the fact that the additive inverse is unique.) 5. Let z=2+3i, w=1—i. Write®z+w,zw,\z\,limtheforma—l-bz’. 6. Solve 3:2 ~4m+ 13 == 0 in C. 7. Describe the plane in R3 through (1 2,3), (2,0,1), (0,1 0). (w+4§) 8. (extra credit) Let 3:, y E Z Let a: N y if and only if fily +432. Prove 1311 t N is an uivalence relation ’F 3 eq ANSMQ, $3 5 UH — Math 4377/6308 - Dr. Heier - Fall 2012 HW 1 Due 09/05, at the beginning of class. Use regular sheets of paper, stapled together. Don’t forget to write your name on page 1. 1. (1 point) Let A = {1,2,5}, B = {4,5}, 0 = {4, 6}. Explicitly write down the sets 5‘53 AuB, An(BUC‘), Bn(A\B), AxO’. 2. (3 points) Let 3:, y E Z. Prove or disprove that the following relations are equivalence relations. (a)w~yifandonlyifm—yisl$sthan10. (b) a:~yifendon1yifm-y20. (c) x~yifandonlyifx~yiseven @(1 point) Give an example of a set A and an relation on A which is symmetric and transitive, but not reflexive. You must not use the example of the empty relation, which has the required properties, but makes this appear more mysterious than it actually is. Hint: There is a. very neat and simple one for A = Z. 4. (3 points) Let f : {0, 1,2,3,4} -~> N,n H n3 + n. (8.) Find the domain, codomain and range of f. (b) Is f one-to-one? (o) Is f onto? (1 point) Give an example of a, real interval I on which the standard sin function is one-to-one with the additional property that sin is not one-to-one on any set strictly containing I . Explain your answer carefully, assuming standard facts about sin without proof. :t 1 l 6. (0.5 points) Let 2 = 1+ 31', w = l - 22'. Write 2, z + w, zw, M, i in the form a+ bi. E055 7. (0.5 points) Solve 22 — 4z + 13 = 0 in C. E“, 8. (1 extra credit point) Let :1:,y E Z. Let 9: ~ y if and only if y + 4a: is an integer multiple of 5. Prove that N is an equivalence relation. ...
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