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

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{n},$

is injective and/or surjective.

Exercise

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

Exercise

Give examples of mappings

$\varphi ,\psi \colon \mathbb {N} \longrightarrow \mathbb {N} ,$

such that ${}\varphi$ is injective, but not surjective, and ${}\psi$ is surjective, but not injective.

Exercise

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

$F\colon L\longrightarrow M$

be a function. Let

$G\colon M\longrightarrow L$

be another function such that ${}F\circ G=\operatorname {Id} _{M}\,$ and ${}G\circ F=\operatorname {Id} _{L}\,$. Show that ${}G$ is the inverse of ${}F$.

Exercise

Determine the composite functions ${}\varphi \circ \psi$ and ${}\psi \circ \varphi$ for the functions ${}\varphi ,\psi \colon \mathbb {R} \rightarrow \mathbb {R}$, defined by

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

Exercise

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

$F\colon L\longrightarrow M,x\longmapsto F(x),$

$G\colon M\longrightarrow N,y\longmapsto G(y),$

and

$H\colon N\longrightarrow P,z\longmapsto H(z),$

be functions. Show that

${}H\circ (G\circ F)=(H\circ G)\circ F\,.$

Exercise *

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

$f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N$

be functions with their composition

$g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).$

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

Exercise

Let

$f_{1},\ldots ,f_{n}\colon \mathbb {R} \longrightarrow \mathbb {R}$

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 ${}\mathbb {C} [X]$ 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. ${}\operatorname {deg} \,(P+Q)\leq \max\{\operatorname {deg} \,(P),\,\operatorname {deg} \,(Q)\}\,,$

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

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 ${}PQ\neq 0$.

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

$\psi \colon K[X]\longrightarrow K,P\longmapsto P(a),$

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 ${}P\in K[X]$, ${}P\neq 0$, can be decomposed as a product

${}P=(X-\lambda _{1})^{\mu _{1}}\cdots (X-\lambda _{k})^{\mu _{k}}\cdot Q\,$

where ${}\mu _{j}\geq 1$ and ${}Q$ is a polynomial with no roots (no zeroes). Moreover, the different numbers ${}\lambda _{1},\ldots ,\lambda _{k}$ and the exponents ${}\mu _{1},\ldots ,\mu _{k}$ are uniquely determined apart from the order.

Exercise

Let ${}F\in \mathbb {C} [X]$ 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

$f=g/h\colon U\longrightarrow \mathbb {R} ,$

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

$\varphi \colon M\longrightarrow M,x\longmapsto \varphi (x),$

defined by the following table

${}x$	${}1$	${}2$	${}3$	${}4$	${}5$	${}6$	${}7$	${}8$
${}\varphi (x)$	${}2$	${}5$	${}6$	${}1$	${}4$	${}3$	${}7$	${}7$

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

Exercise (2 marks)

Prove that a strictly increasing function

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

is injective.

Exercise (3 marks)

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

$f:L\longrightarrow M{\text{ and }}g:M\longrightarrow N$

be functions with their composite

$g\circ f\colon L\longrightarrow N,x\longmapsto g(f(x)).$

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

Exercise (3 marks)

Compute in the polynomial ring ${}\mathbb {C} [X]$ 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 ${}\mathbb {C} [X]$ 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 ${}P\in \mathbb {R} [X]$ be a non-constant polynomial with real coefficients. Prove that ${}P$ can be written as a product of real polynomials of degrees ${}1$ or ${}2$.