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BIBLIOGRAPHY AND

REFERENCES

Proofs in Mathematics

1.Bloch, Ethan D., Proofs and Fundamentals: A First Course in Abstract Mathematics. Boston: Birkhauser, 2000.

2.Schumacher, Carol, Chapter Zero: Fundamental Notions of Abstract Mathematics, 2nd ed. Reading, Mass.: Addison-Wesley, 2000.

3.Solow, Daniel, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, 3rd ed. New York: Wiley, 2002.

Modern Algebra in General

4.Artin, Michael, Algebra. Upper Saddle River, N.J.: Prentice Hall, 1991.

5.Birkhoff, Garrett, and Thomas C. Bartee, Modern Applied Algebra. New York: McGraw-Hill, 1970.

6.Birkhoff, Garrett, and Saunders Maclane, A Survey of Modern Algebra, 4th ed. New York: Macmillan, 1977.

7.Durbin, John R., Modern Algebra: An Introduction, 4th ed. New York: Wiley, 2000.

8.Gallian, Joseph A., Contemporary Abstract Algebra, 5th ed. Boston: Houghton Mifflin, 2002.

9.Herstein, I. N., Topics in Algebra, 2nd ed. New York: Wiley, 1973.

10.Lidl, Rudolf, and Gunter Pilz, Applied Abstract Algebra, 2nd ed. New York: Springer-Verlag, 1997.

11.Nicholson, W. Keith, Introduction to Abstract Algebra, 2nd ed. New York: Wiley, 1999.

12.Weiss, Edwin, First Course in Algebra and Number Theory. San Diego, Calif.: Academic Press, 1971.

History of Modern Algebra

13.Kline, Morris, Mathematical Thought from Ancient to Modern Times, Vol. 3. New York: Oxford University Press, 1990 (Chap. 49).

14.Stillwell, John, Mathematics and Its History, 2nd ed. New York: Springer-Verlag, 2002.

Modern Algebra with Applications, Second Edition, by William J. Gilbert and W. Keith Nicholson ISBN 0-471-41451-4 Copyright 2004 John Wiley & Sons, Inc.

BIBLIOGRAPHY AND REFERENCES

307

Connections to Computer Science and Combinatorics

15.Biggs, Norman L., Discrete Mathematics, 2nd ed. Oxford: Oxford University Press, 2003.

16.Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, 2nd ed. Cambridge: Cambridge University Press, 2002.

17.Gathen, Joachim von zur, and Jurgen¨ Gerhard, Modern Computer Algebra, 2nd ed. Cambridge: Cambridge University Press, 2003.

18.Hopcroft, John E., Rajeev Motwani, and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation, 2nd ed. Reading, Mass.: Addison-Wesley, 2000.

19.Knuth, Donald E., The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, 3rd ed. Reading, Mass.: Addison-Wesley, 1998.

20.Kolman, Bernard, Robert C. Busby, and Sharon Cutler Ross, Discrete Mathematical Structures, 4th ed. Upper Saddle River, N.J.: Prentice Hall, 1999.

21.Mendelson, Elliott, Schaum’s Outline of Theory and Problems of Boolean Algebra and Switching Circuits. New York: McGraw-Hill, 1970.

22.Stone, Harold S., Discrete Mathematical Structures and Their Applications. Chicago: Science Research Associates, 1973.

23.Whitesitt, J. Eldon, Boolean Algebra and Its Applications. New York: Dover, 1995.

Groups and Symmetry

24.Armstrong, Mark Anthony, Groups and Symmetry. New York: Springer-Verlag, 1988.

25.Baumslag, Benjamin, and Bruce Chandler, Schaum’s Outline of Group Theory. New York: McGraw-Hill, 1968.

26.Budden, F. J., The Fascination of Groups. Cambridge: Cambridge University Press, 1972.

27.Coxeter, H. S. M., Introduction to Geometry, 2nd ed. New York: Wiley, 1989.

28.Cundy, H. Martyn, and A. P. Rollett, Mathematical Models, 3rd ed. Stradbroke, Norfolk, England: Tarquin, 1981.

29.Field, Michael, and Martin Golubitsky, Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature. Oxford: Oxford University Press, 1992.

30.Hall, Marshall, Jr., The Theory of Groups. New York: Macmillan, 1959 (reprinted by the American Mathematical Society, 1999).

31.Lomont, John S., Applications of Finite Groups. New York: Dover, 1993.

32.Shapiro, Louis W., Finite groups acting on sets with applications. Mathematics Magazine, 46 (1973), 136–147.

Rings and Fields

33.Cohn, P. M., Introduction to Ring Theory. New York: Springer-Verlag, 2000.

34.Lidl, Rudolf, and Harald Niederreiter, Introduction to Finite Fields and Their Applications, rev. ed. Cambridge: Cambridge University Press, 1994.

35.Stewart, Ian, Galois Theory, 3rd ed. Boca Raton, Fla.: CRC Press, 2003.

Convolution Fractions

36.Erdelyi, Arthur, Operational Calculus and Generalized Functions. New York: Holt, Rinehart and Winston, 1962.

37.Marchand, Jean Paul, Distributions: An Outline. Amsterdam: North-Holland, 1962.

Latin Squares

38.Ball, W. W. Rouse, and H. S. M. Coxeter, Mathematical Recreations and Essays. New York: Dover, 1987.

308

BIBLIOGRAPHY AND REFERENCES

39.Lam, C. W. H., The search for a finite projective plane of order 10. American Mathematical Monthly, 98(1991), 305–318.

40.Laywine, Charles F., and Gary L. Mullen, Discrete Mathematics Using Latin Squares. New York: Wiley, 1998.

Geometrical Constructions

41.Courant, Richard, Herbert Robbins, and Ian Stewart, What Is Mathematics? New York: Oxford University Press, 1996.

42.Kalmanson, Kenneth, A familiar constructibility criterion. American Mathematical Monthly, 79(1972), 277–278.

43.Kazarinoff, Nicholas D., Ruler and the Round. New York: Dover, 2003.

44.Klein, Felix, Famous Problems of Elementary Geometry. New York: Dover, 1956.

Coding Theory

45.Kirtland, Joseph, Identification Numbers and Check Digit Schemes. Washington, D.C.: Mathematical Association of America, 2001.

46.Roman, Steven, Introduction to Coding and Information Theory. New York: Springer-Verlag, 1997.

ANSWERS TO THE

ODD-NUMBERED

EXERCISES

CHAPTER 2

 

2.1. Always true.

2.3. When A (B C) = .

2.5. When A (B C) = .

 

2.13.|A B C D| = |A| + |B| + |C| + |D| − |A B| −|A C| − |A D| − |B C| − |B D| − |C D|

+|A B C| + |A B D| + |A C D| + |B C D|

−|A B C D|.

 

 

2.17. Yes; P( ).

2.15. 4.

 

 

 

 

 

2.25.

 

 

 

 

 

 

 

A

B

(a)

(b)

(c)

(d)

 

 

 

 

 

 

 

 

 

 

T

T

T

T

T

T

 

T

F

F

F

T

T

 

F

T

T

T

F

F

 

F

F

T

T

T

T

 

 

 

 

 

 

 

 

(a) and (b) are equivalent and (c) and (d) are equivalent.

2.27.(a) is a contradiction and (b), (c) and (d) are tautologies.

2.29.

B A C

D

BC

B

2.31.B C .

2.33. A (B A ); (A B) (A B ) (A B); A B.

Modern Algebra with Applications, Second Edition, by William J. Gilbert and W. Keith Nicholson ISBN 0-471-41451-4 Copyright 2004 John Wiley & Sons, Inc.