Homework Assignment 1МАTH 541This homework assignment corresponds with material covered through January 29, Sections 0.1, 0.2, 0.38 1.1 in Dummit & Foote Abstract Algebra 3rd Ed. Please write the problems and solutions on separatesheets of paper. The calculations, erplanations, and proofs should be in complete coherent and logicalsentences. This assignment is due on Thursday, February 4 at 11:59pm in Canvas as a .pdf file.1. Exercise 5 in 80.1. Determine whether the following functions f are well defined:(a) f:Q→ Z defined by f(a/b) = a.(b) f:Q→Q defined by f(a/b) = a² /b°.2. Exercise 7 in $0.1 Let f : A → B be a surjective map of sets. Prove that the relationa - b if and only if f(a) = f(b)is an equivalence relation whose equivalence classes are the fibers of f.Hint: Use Proposition 2 in the textbook.3. Let n > 2. For any a, b e Z, proof that n | (b – a) if and only if a and b have the same remainderwhen divided by n. Conclude that a = b (mod n) if and only if a and b have the same remainderwhen divided by n.4. Exercise 11 in 80.3 Prove that if ā, b € (Z/nZ)*, then ā - be (Z/nZ) *.5. Exercise 6 in §1.1. Determine which of the following sets are groups under addition. Explain why orwhy not.(a) the set of rational numbers (including 0 = 0/1) in lowest terms whose denominators are odd.(b) the set of rational numbers in lowest terms whose denominators are even together with 0.(c) the set of rational numbers of absolute value < 1.(d) the set of rational numbers of absolute value >1 together with 0.(e) the set of rational numbers with denominators equal to 1 or 2.(f) the set of rational numbers with denominators equal to 1, 2, or 3.{(: ) -6. Prove that GLn(R) =: ad – bc + CC Mat2x2(R) (called the general linear group ofdegree 2) is a group under matrix multiplication. Does your argument work for any degree n?Hint: Use the property det(AB) = det A · det B. for all A, B E Matnxn(R).7. Is the set U =: ad – bc = 1C Mat2x2 (R) a group under matrix multiplication?8. Exercise 5 in §1.1. Prove for all n > 1 that Z/nZ is not a group under multiplication of residueclasses.

Answered: Image /qna-images/answer/2c08699b-87a7-4777-906a-69014c818ab4.jpg

2. Exercise 7 in §0.1 Let f : A B be a surjective map of sets. Prove that the relation a-bif and only if f(a) = f(b) is an equivalence relation whose equivalence classes are the fibers of f. Hint: Use Proposition 2 in the textbook.

2. Exercise 7 in §0.1 Let f : A B be a surjective map of sets. Prove that the relation a-bif and only if f(a) = f(b) is an equivalence relation whose equivalence classes are the fibers of f. Hint: Use Proposition 2 in the textbook.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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