Mathematics 3101

Algebraic Structures with Computer Applications

 
Course Outline
 
Notice Board

My office hours during the exam period. Dec. 16 (Wednesday): 11:30am - 1:00am. Dec. 18 (Friday): 11:00pm -12:30pm.

Topics covered in the class.

  • Sept. 10: Introduction; Sets, mappings.
  • Sept. 15: Mappings, Binary operations, Binary relation.
  • Sept. 17: Binary relation, divisibility, greatest common divisor
  • Sept. 22: Euclidean algorithm, unique factorization, congruences.
  • Sept. 24: solve linear congruence, Chinese remainder theorem, congruence classes.
  • Sept. 29. Congruence classes, RSA.
  • Oct. 1. Monoids.
  • Oct. 6. Deterministic Finite Automata, recognizable language, regular language.
  • Oct. 8. Regular language and monoids of row monomial matrices.
  • Oct. 13. Groups, examples and properties.
  • Oct. 15. proof of properties, subgroups, exponents and multiples.
  • Oct. 20. subgroups, cyclic groups.
  • Oct. 22. generators of cyclic groups, Isomorphisms.
  • Oct. 27: Homomorphisms, permutation groups.
  • Oct. 29: permutation groups, Cayley's theorem.
  • Nov. 3: Cosets, normal subgroups.
  • Nov. 5: Normal subgroups, Quotient groups.
  • Nov. 10: Fundamental theorem of homomorphisms.
  • Nov. 12: Second/third isomorphic theorem, rings and fields.
  • NOv. 17: properties of rings, integral domains and fields, units.
  • Nov. 19: ideals and quotient rings.
  • Nov. 24: Maximal ideals and fields, polynomials.
  • Nov. 26: division algorithms, gcd.
  • Dec. 1: irreducible polynomials, constructions of finite fields.
  • Dec. 3: review.

    Tutorials

  • Tutotial #1: Section 1.1: 19 Section 1.2 : 2, 4(a)(d)(e) Section 1.3: 4, 13 Section 2.3: 3, 7, 13, 26 Section 2.4: 3(c)(d).
  • Tutotial #2: Section 2.5: 15, 23, 28, 29(l), 39, 53 (a)(e) Section 2.6: 5 (a)(b), 7(a)(b), 8(a)(b), 16.
  • Tutotial #3: See WebCT.
  • Tutorial #4: Section 3.3. 1,15, 17 Section 3.4: 14, 16, 18, 21, 37 Section 3.5: 7, 10, 16
  • Tutorial #5: Section 4.1: 1, 3, 5, 9, 10, 11 Section 4.2: 8. Section 4.4: 3, 9, 11, 13, 21, 22. Section 4.5: 16-20. Section 4.6: 14, 17.
  • Tutorial #6: Section 5.1: 21. Section 5.2: 11. Section 6.1: 4.
  • Tutorial #7: Section 8.2: 1, 7, 9. Section 8.3: 4. Section 8.6: 2.

    More suggested exercises:

  • 1.1: 1-15.
  • 1.2: 1-10.
  • 1.4: 1-13.
  • 1.7: 1-5, 11-12.
  • 2.3: 1-3, 6-7, 17-18, 24-27.
  • 2.4: 2-3, 7-14, 18-25, 29-30, 39-45, 53.
  • 2.5: 1- 50, 53.
  • 2.6: 1-14.

    Test #1 covers the above topics

  • 2.8: 17-22. (not section 2.7!)
  • 3.1: 1-20: decide whether it is a monoid or not.
  • Automata (tutorial #3)
  • 3.1: 1-35.
  • 3.2: 1-22.

    Test #2 covers (section 2.7, monoids and automata theory, sections 3.1, 3.2).

  • 3.3: 1-29;
  • 3.4: 1-29;
  • 3.5: 1-20.
  • 3.6: 1-21.
  • 4.1: 1-20.
  • 4.2: 1-6, 8-10.

    Test #3 covers up to here.

  • 4.4: 3, 5, 7-13.
  • 4.5: 1-7.
  • 4.6: 1-9, 13-26, 28-29, 36.
  • 5.1: 1-2, 7-8, 11-40.
  • 5.2: 1, 11, 15, 20, 23-24.

    Test #4 covers up to here.

  • 6.1: 1, 4, 6, 7, 8, 24.
  • 6.2: 6, 8, 9, 10.
  • 6.4: 3, 4, 9, 10.
  • 8.2: 1-14, 19-22.
  • 8.3: 4.
  • 8.6: 1-6.