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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—lbz’. 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:~yifendon1yifmy20.
(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 onetoone? (o) Is f onto? (1 point) Give an example of a, real interval I on which the standard sin function
is onetoone with the additional property that sin is not onetoone 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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 Fall '08
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 Linear Algebra, Algebra, Sets

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