Generators and Direct Products Homework Section A

(1) Verify the equalities for subgroups of : and ]]> http://www.libraryofmath.com/cayley-theorem-homework.html http://www.libraryofmath.com/cayley-theorem-homework.html Thu, 22 May 2008 15:42:00 -0400

Cayley's Theorem Homework Section A

(1) Write the permutation associated with each element of by the isomorphism in the proof of Cayley's theorem. ]]> http://www.libraryofmath.com/more-on-isomorphism-homework.html http://www.libraryofmath.com/more-on-isomorphism-homework.html Thu, 22 May 2008 15:42:00 -0400

More on Isomorphism Homework Section A

(1) Determine if the groups are isomorphic or not: and ]]> http://www.libraryofmath.com/isomorphism-homework.html http://www.libraryofmath.com/isomorphism-homework.html Thu, 22 May 2008 15:42:00 -0400

Isomorphism Homework Section A

(1) Prove that is isomorphic to the multiplicative group of all rational numbers of the form for ]]> http://www.libraryofmath.com/lagrange-theorem-homework.html http://www.libraryofmath.com/lagrange-theorem-homework.html Thu, 22 May 2008 15:42:00 -0400

Lagrange's Theorem Homework Section A

(1) Find ]]> http://www.libraryofmath.com/cosets-homework.html http://www.libraryofmath.com/cosets-homework.html Thu, 22 May 2008 15:42:00 -0400

Cosets Homework Section A

(1) Determine the right cosets of in ]]> http://www.libraryofmath.com/elementary-properties-of-groups-homework.html http://www.libraryofmath.com/elementary-properties-of-groups-homework.html Thu, 22 May 2008 15:42:00 -0400

Elementary Properties of Groups Homework Section A

(1) Determine the elements in each of the cyclic subgroups of Also give the order of each element of ]]> http://www.libraryofmath.com/permutations-homework.html http://www.libraryofmath.com/permutations-homework.html Thu, 22 May 2008 15:42:00 -0400

Permutations Homework Section A

(1) Assume and compute and ]]> http://www.libraryofmath.com/group-definitions-and-examples-homework.html http://www.libraryofmath.com/group-definitions-and-examples-homework.html Thu, 22 May 2008 15:42:00 -0400

Group Definitions and Examples Homework Section A

(1) Show that is a group with respect to multiplication. ]]> http://www.libraryofmath.com/subgroups-homework.html http://www.libraryofmath.com/subgroups-homework.html Thu, 22 May 2008 15:42:00 -0400

Subgroups Homework Section A

(1) Decide whether the given subset is a subgroup of

    (a)
    
    (b)
    
    (c)
    
    (d)
     ]]> http://www.libraryofmath.com/groups-and-symmetry-homework.html http://www.libraryofmath.com/groups-and-symmetry-homework.html Thu, 22 May 2008 15:42:00 -0400

Groups and Symmetry Homework Section A

(1) Determine the group of symmetries of a square. ]]> http://www.libraryofmath.com/equivalence-relations-homework.html http://www.libraryofmath.com/equivalence-relations-homework.html Thu, 22 May 2008 15:42:00 -0400

Equivalence Relations Homework Section A

(1) For point and in a plane with rectangular coordinate system, let mean that Either show is an equivalence relation on or show that it is not. ]]> http://www.libraryofmath.com/operations-homework.html http://www.libraryofmath.com/operations-homework.html Thu, 22 May 2008 15:42:01 -0400

Operations Homework Section A

(1) Does define an operation on the set of all integers? ]]> http://www.libraryofmath.com/composition-as-an-operation-homework.html http://www.libraryofmath.com/composition-as-an-operation-homework.html Thu, 22 May 2008 15:42:01 -0400

Composition as an Operation Homework Section A

(1) With the set contains four elements, denote these by and defined as follows



Construct the Cayley table for composition as an operation on Which is the identity element? Is commutative as an operation on Which elements of are invertible? Is commutative as an operation on the set of invertible elements in ]]> http://www.libraryofmath.com/composition-and-invertible-mappings-homework.html http://www.libraryofmath.com/composition-and-invertible-mappings-homework.html Thu, 22 May 2008 15:42:01 -0400

Composition and Invertible Mappings Homework Section A

(1) Let and be mappings from to Write a formula for and In each case determine the image. ]]> http://www.libraryofmath.com/mappings-homework.html http://www.libraryofmath.com/mappings-homework.html Thu, 22 May 2008 15:42:01 -0400

Mappings Homework Section A

(1) Let and be mappings from to defined by and for each Which of these mappings are onto? Which of these mappings are one-to-one? Determine the range of   and ]]> http://www.libraryofmath.com/abstract-algebra-one.html http://www.libraryofmath.com/abstract-algebra-one.html Thu, 22 May 2008 15:42:01 -0400

about-abstract-algebra-one
abstract-algebra-one-homework
abstract algebra one learning

]]> http://www.libraryofmath.com/about-mathematics.html http://www.libraryofmath.com/about-mathematics.html Thu, 22 May 2008 15:42:01 -0400 The mathematical method is a process of doing logically deduction and abstract reasoning involving the relationships of quantities and sets which has evolved through the use of abstraction and analytical reasoning. To do mathematics is to explore concepts and formulate new conjectures and establish their validity by rigorous deduction from agreed upon axioms and definitions. Mathematics is a significant ingredient of our culture and we are willing to learn more about it. ]]> http://www.libraryofmath.com/zeros-of-polynomials-quiz.html http://www.libraryofmath.com/zeros-of-polynomials-quiz.html Thu, 22 May 2008 15:42:01 -0400

Show all work and justify each step.

(1) Find the zeros of and state the multiplicity of each. Sketch the graph of each function

    (a)
    
    (b)

(2) Find a polynomial of degree 7 such that and are both zeros of multiplicity 2, 0 is a zero of multiplicity 3, and Sketch the graph of

(3) Construct a polynomial function with the stated properties.

    (a) Fifth degree, only real coefficients, 0 is the only real zero, is a zero of multiplicity 1, leading coefficient is 1.
    
    (b) Fourth degree, only real coefficients, -intercepts are 0 and 6, is a zero, leading coefficient is 3.
    
    (c) Fifth degree, is a zero of multiplicity 2, another integer is a zero of multiplicity 3, and -intercept is leading coefficient is 1.
]]> http://www.libraryofmath.com/zeros-of-polynomials.html http://www.libraryofmath.com/zeros-of-polynomials.html Thu, 22 May 2008 15:42:01 -0400 Example (Zeros of Polynomials) Find a polynomial od degree 3 that has the zeros and Sketch the graph of

    Solution. ]]> http://www.libraryofmath.com/zeros-of-a-polynomial.html http://www.libraryofmath.com/zeros-of-a-polynomial.html Thu, 22 May 2008 15:42:01 -0400 Given a polynomial function of degree there are some obvious questions that can be difficult to answer.

    (i) How many zeros of are real and how many are complex?
    
    (ii) How many zeros are positive and how many are negative?
    
    (iii) Ho many zeros are rational and how many are irrational?
    
The following famous theorem is called The Fundamental  of Algebra and give is a good starting point for these questions. ]]> http://www.libraryofmath.com/zero-derivative-theorem.html http://www.libraryofmath.com/zero-derivative-theorem.html Thu, 22 May 2008 15:42:01 -0400     The following theorem is a partial converse to the statement that the derivative of a constant is 0. ]]> http://www.libraryofmath.com/work-as-a-line-integral.html http://www.libraryofmath.com/work-as-a-line-integral.html Thu, 22 May 2008 15:42:01 -0400 In this topic:

    (1) State the relationship between a line integral and the work performed as an object moves along a smooth curve .
    
    (2) Find the work done by the force field on an object moving along the curve defined parametrically by for
     ]]> http://www.libraryofmath.com/wilsons-theorem-and-fermats-little-theorem-homework-key.html http://www.libraryofmath.com/wilsons-theorem-and-fermats-little-theorem-homework-key.html Thu, 22 May 2008 15:42:01 -0400 Directions: Write legibly and in pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution, show all steps, and write the final answer in a sentence. Do not turn in your scratch work. Staple your pages together, in the correct order, and use this page as a cover sheet. ]]>