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Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 4

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Warm-up-exercises

Establish, for each , whether the function

is injective and/or surjective.


Show that there exists a bijection between and .


Give examples of mappings

such that is injective but not surjective, and is surjective but not injective.


Let and be sets and let

be a function. Let

be another function such that and . Show that is the inverse of .


Determine the composite functions and for the functions , defined by


Let and be sets and let

and

be functions. Show that


Let be sets and let

be mappings with their composition

Show that if is injective, then also is injective.


Let

be functions, which are increasing or decreasing, and let be their composition. Let be the number of the decreasing functions among the 's. Show that if is even, then is increasing, and if is odd, then is decreasing.


Calculate in the polynomial ring the product


Let be a field and let be the polynomial ring over . Prove the following properties concerning the degree of a polynomial:


Show that in a polynomial ring over a field , the following statement holds: if are not zero, then also .


Let be a field and let be the polynomial ring over . Let . Prove that the evaluating function

satisfies the following properties (here let ).


Evaluate the polynomial

replacing the variable by the complex number .


Perform, in the polynomial ring , the division with remainder , where , and .


Let be a field and let be the polynomial ring over . Show that every polynomial , , can be decomposed as a product

where and is a polynomial with no roots (no zeroes). Moreover, the different numbers and the exponents are uniquely determined apart from the order.


Let be a non-constant polynomial. Prove that can be decomposed as a product of linear factors.


Determine the smallest real number for which the Bernoulli inequality with exponent holds.


Sketch the graph of the following rational functions

where each time is the complement set of the set of the zeros of the denominator polynomial .

  1. ,
  2. ,
  3. ,
  4. ,
  5. ,
  6. ,
  7. .


Let be a polynomial with real coefficients and let be a root of . Show that also the complex conjugate is a root of .




Hand-in-exercises

Exercise (3 marks)

Consider the set , and the mapping

defined by the following table

Compute , that is, the -rd composition (or iteration) of with itself.


Exercise (2 marks)

Prove that a strictly increasing function

is injective.


Exercise (3 marks)

Let be sets, and let

be mappings with their composite mapping

Show that if is surjective, then also is surjective.


Exercise (3 marks)

Compute in the polynomial ring the product


Exercise (4 marks)

Perform, in the polynomial ring , the division with remainder , where

and


Exercise (4 marks)

Let be a non-constant polynomial with real coefficients. Prove that can be written as a product of real polynomials of degrees or .