Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 4/en

Aus Wikiversity
Zur Navigation springen Zur Suche springen



Warm-up-exercises

Exercise

Establish for each if the function

is injective

and/or surjective.


Exercise

Show that there exists a bijection between and .


Exercise

Give examples of functions

such that is

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


Exercise

Let and be two sets and let

be a function. Let

be another function such that

and . Show that is the inverse of .


Exercise

Determine the composite functions

for the functions , defined by


Exercise

Let and be sets and let

and
be

functions. Show that


Exercise

Let be sets and let

be functions with their composition

Show that if is injective, then also is injective.


Exercise

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.


Exercise

Calculate in the polynomial ring the product


Exercise

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

  1. ,
  2. .


Exercise

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


Exercise

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

satisfies the following properties (here let ).

  1. .
  2. .
  3. .


Exercise

Evaluate the polynomial

replacing the variable by the

complex number .


Exercise

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


Exercise

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.


Exercise

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


Exercise

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


Exercise

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. .


Exercise

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 points)

Consider the set and the function

defined by the following table

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


Exercise (2 points)

Prove that a strictly increasing function

is injective.


Exercise (3 points)

Let be sets and let

be functions with their composite

Show that if is surjective, then also is surjective.


Exercise (3 points)

Calculate in the polynomial ring the product


Exercise (3 points)

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


Exercise (5 points)

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



<< | Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I | >>

PDF-Version dieses Arbeitsblattes (PDF englisch)

Zur Vorlesung (PDF)