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

Warm-up-exercises

Exercise

Establish, for each $n\in \mathbb{N}$, whether the function

$ℝ⟶ℝ,x⟼x^n,$

is injective and/or surjective.

Exercise

Show that there exists a bijection between $\mathbb{N}$ and $\mathbb{Z}$.

Exercise

Give examples of mappings

$\phi ,\psi :\mathbb{N}⟶\mathbb{N}$

such that $\phi$ is injective but not surjective, and $\psi$ is surjective but not injective.

Exercise

Let $L$ and $M$ be sets and let

$F:L⟶M$

be a function. Let

$G:M⟶L$

be another function such that $F\circ G={\mathrm{Id}}_M$ and $G\circ F={\mathrm{Id}}_L$. Show that $G$ is the inverse of $F$.

Exercise

Determine the composite functions $\phi \circ \psi$ and $\psi \circ \phi$ for the functions $\phi ,\psi :ℝ\to ℝ$, defined by

$\phi (x)=x^4+3x^2-2x+5\text{ and }\psi (x)=2x^3-x^2+6x-1.$

Exercise *

Let $L,M,N$ and $P$ be sets and let

$F:L⟶M,x⟼F(x),$

$G:M⟶N,y⟼G(y),$

and

$H:N⟶P,z⟼H(z),$

be functions. Show that

$H\circ (G\circ F)=(H\circ G)\circ F.$

Exercise *

Let $L,M,N$ be sets and let

$f:L⟶M\text{ and }g:M⟶N$

be mappings with their composition

$g\circ f:L⟶N,x⟼g(f(x)).$

Show that if $g\circ f$ is injective, then also $f$ is injective.

Exercise

Let

$f_1,\dots ,f_n:ℝ⟶ℝ$

be functions, which are increasing or decreasing, and let $f=f_n\circ \cdots \circ f_1$ be their composition. Let $k$ be the number of the decreasing functions among the $f_i$'s. Show that if $k$ is even, then $f$ is increasing, and if $k$ is odd, then $f$ is decreasing.

Exercise

Calculate in the polynomial ring $ℂ[X]$ the product

$((4+\mathrm{i})X^2-3X+9\mathrm{i})\cdot ((-3+7\mathrm{i})X^2+(2+2\mathrm{i})X-1+6\mathrm{i}).$

Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Prove the following properties concerning the degree of a polynomial:

1. $\mathrm{deg}(P+Q)\le max\{\mathrm{deg}(P),\mathrm{deg}(Q)\},$

2. $\mathrm{deg}(P\cdot Q)=\mathrm{deg}(P)+\mathrm{deg}(Q).$

Exercise

Show that in a polynomial ring over a field $K$, the following statement holds: if $P,Q\in K[X]$ are not zero, then also $PQ\ne 0$.

Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let $a\in K$. Prove that the evaluating function

$\psi :K[X]⟶K,P⟼P(a),$

satisfies the following properties (here let $P,Q\in K[X]$).

1. $(P+Q)(a)=P(a)+Q(a).$

2. $(P\cdot Q)(a)=P(a)\cdot Q(a).$

3. $1(a)=1.$

Exercise

Evaluate the polynomial

$2X^3-5X^2-4X+7$

replacing the variable $X$ by the complex number $2-5\mathrm{i}$.

Exercise

Perform, in the polynomial ring $\mathbb{Q}[X]$, the division with remainder $\frac{P}{T}$, where $P=3X^4+7X^2-2X+5$, and $T=2X^2+3X-1$.

Exercise

Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Show that every polynomial $P\in K[X]$, $P\ne 0$, can be decomposed as a product

$P=(X-{\lambda }_1{)}^{{\mu }_1}\cdots (X-{\lambda }_k{)}^{{\mu }_k}\cdot Q$

where ${\mu }_j\ge 1$ and $Q$ is a polynomial with no roots (no zeroes). Moreover, the different numbers ${\lambda }_1,\dots ,{\lambda }_k$ and the exponents ${\mu }_1,\dots ,{\mu }_k$ are uniquely determined apart from the order.

Exercise

Let $F\in ℂ[X]$ be a non-constant polynomial. Prove that $F$ can be decomposed as a product of linear factors.

Exercise

Determine the smallest real number for which the Bernoulli inequality with exponent $n=3$ holds.

Exercise

Sketch the graph of the following rational functions

$f=g/h:U⟶ℝ,$

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

1. $1/x$,
2. $1/x^2$,
3. $1/(x^2+1)$,
4. $x/(x^2+1)$,
5. $x^2/(x^2+1)$,
6. $x^3/(x^2+1)$,
7. $(x-2)(x+2)(x+4)/(x-1)x(x+1)$.

Exercise

Let $P\in ℝ[X]$ be a polynomial with real coefficients and let $z\in ℂ$ be a root of $P$. Show that also the complex conjugate $\bar{z}$ is a root of $P$.

Hand-in-exercises

Exercise (3 marks)

Consider the set $M=\{1,2,3,4,5,6,7,8\}$, and the mapping

$\phi :M⟶M,x⟼\phi (x),$

defined by the following table

$x$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$\phi (x)$	$2$	$5$	$6$	$1$	$4$	$3$	$7$	$7$

Compute ${\phi }^{1003}$, that is, the $1003$-rd composition (or iteration) of $\phi$ with itself.

Exercise (2 marks)

Prove that a strictly increasing function

$f:ℝ⟶ℝ$

is injective.

Exercise (3 marks)

Let $L,M,N$ be sets, and let

$f:L⟶M\text{ and }g:M⟶N$

be mappings with their composite mapping

$g\circ f:L⟶N,x⟼g(f(x)).$

Show that if $g\circ f$ is surjective, then also $g$ is surjective.

Exercise (3 marks)

Compute in the polynomial ring $ℂ[X]$ the product

$((4+\mathrm{i})X^3-\mathrm{i}{X}^2+2X+3+2\mathrm{i})\cdot ((2-\mathrm{i})X^3+(3-5\mathrm{i})X^2+(2+\mathrm{i})X+1+5\mathrm{i}).$

Exercise (4 marks)

Perform, in the polynomial ring $ℂ[X]$, the division with remainder $\frac{P}{T}$, where

$P=(5+X^2+\mathrm{i}X+3-\mathrm{i})X^4+X^2+\mathrm{i}X+3-\mathrm{i}{X}^2+(3-2X^2+\mathrm{i}X+3-\mathrm{i})X-1$

and

$T=X^2+\mathrm{i}X+3-\mathrm{i}.$

Exercise (4 marks)

Let $P\in ℝ[X]$ be a non-constant polynomial with real coefficients. Prove that $P$ can be written as a product of real polynomials of degrees $1$ or $2$.