Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 4/en/latex
\setcounter{section}{4}
\zwischenueberschrift{Warm-up-exercises}
\inputexercise
{}
{
Establish for each $n \in \N$ if the function \maabbeledisp {} {\R} {\R } {x} {x^n } {,} is \definitionsverweis {injective}{}{} and/or \definitionsverweis {surjective}{}{.}
}
{} {}
\inputexercise
{}
{
Show that there exists a \definitionsverweis {bijection}{}{} between \mathkor {} {\N} {and} {\Z} {.}
}
{} {}
\inputexercise
{}
{
Give examples of \definitionsverweis {functions}{}{} \maabbdisp {\varphi, \psi} {\N} {\N } {} such that $\varphi$ is \definitionsverweis {injective}{}{,} but not \definitionsverweis {surjective}{}{}, and $\psi$ is surjective, but not injective.
}
{} {}
\inputexercise
{}
{
Let $L$ and $M$ be two sets and let \maabbdisp {F} {L} {M
} {} be a function. Let \maabbdisp {G} {M} {L
} {} be another function such that \mathkor {} {F \circ G = \operatorname{id}_M \,} {and} {G \circ F = \operatorname{id}_L \,} {}. Show that $G$ is the \definitionsverweis {inverse}{}{} of $F$.
}
{} {}
\inputexercise
{}
{
Determine the
\definitionsverweis {composite functions}{}{}
\mathdisp {\varphi \circ \psi \text{ and } \psi \circ \varphi} { }
for the
\definitionsverweis {functions}{}{}
\maabb {\varphi,\psi} {\R} {\R
} {,} defined by
\mathdisp {\varphi(x)=x^4+3x^2-2x+5 \text{ and } \psi(x)=2x^3-x^2+6x-1} { . }
}
{} {}
\inputexercise
{}
{
Let
$L, M, N$ and $P$ be sets and let
\maabbeledisp {F} {L} {M
} {x} {F(x)
} {,} \maabbeledisp {G} {M} {N
} {y} {G(y)
} {,} and
\maabbeledisp {H} {N} {P
} {z} {H(z)
} {,} be
\definitionsverweis {functions}{}{.}
Show that
\mathdisp {\verknuepfung {(\verknuepfung {F} {G})} {H} = \verknuepfung {F} {(\verknuepfung {G} {H})}} { . }
}
{} {}
\inputexercise
{}
{
Let $L,M,N$ be sets and let
\mathdisp {f:L \longrightarrow M \text{ and } g:M \longrightarrow N} { }
be
\definitionsverweis {functions}{}{}
with their
\definitionsverweis {composition}{}{}
\maabbeledisp {\verknuepfung {f} {g}} {L} {N
} {x} {g(f(x))
} {.}
Show that if $\verknuepfung {f} {g}$ is
\definitionsverweis {injective}{}{,}
then also $f$ is injective.
}
{} {}
\inputexercise
{}
{
Let
\maabbdisp {f_1 , \ldots , f_n} {\R} {\R
} {} be functions, which are increasing or decreasing, and let
\mathl{f=f_n \circ \cdots \circ f_1}{} be their
\definitionsverweis {composition}{}{.}
Let $k$ be the number of the decreasing functions among the $f_i$'s. Show that if $k$ is even then $f$ is
\definitionsverweis {increasing}{}{} and if $k$ is odd then $f$ is
\definitionsverweis {decreasing}{}{.}
}
{} {}
\inputexercise
{}
{
Calculate in the
\definitionsverweis {polynomial ring}{}{} ${\mathbb C}[X]$ the product
\mathdisp {((4+i)X^2-3X+9i) \cdot ((-3+7i)X^2+(2+2i)X-1+6i)} { . }
}
{} {}
\inputexercise
{}
{
Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Prove the following properties concerning the \definitionsverweis {degree}{}{} of a polynomial: \aufzaehlungzwei { $\operatorname{deg} \, (P+Q) \leq \max \{ \operatorname{deg} \, (P),\, \operatorname{deg} \, (Q)\}$, } { $\operatorname{deg} \, (P \cdot Q) = \operatorname{deg} \, (P) + \operatorname{deg} \, (Q)$. }
}
{} {}
\inputexercise
{}
{
Show that in a \definitionsverweis {polynomial ring}{}{} over a \definitionsverweis {field}{}{} $K$ the following statement holds: if $P,Q \in K[X]$ are not zero, then also $PQ \neq 0$.
}
{} {}
\inputexercise
{}
{
Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Let
\mathl{a \in K}{.} Prove that the evaluating function
\maabbeledisp {\psi} {K[X]} {K
} {P} {P(a)
} {,}
satisfies the following properties
\zusatzklammer {here let \mathlk{P,Q \in K[X]}{}} {} {.}
\aufzaehlungdrei{$(P + Q)(a)=P(a) +Q(a)$.
}{$(P \cdot Q)(a)=P(a) \cdot Q(a)$.
}{$1(a)=1$.
}
}
{} {}
\inputexercise
{}
{
Evaluate the
\definitionsverweis {polynomial}{}{}
\mathdisp {2X^3-5X^2-4X+7} { }
replacing the variable $X$ by the
\definitionsverweis {complex number}{}{} $2-5i$.
}
{} {}
\inputexercise
{}
{
Perform, in the polynomial ring $\Q[X]$ the division with remainder $\frac{P}{T}$, where $P=3X^4+7X^2-2X+5$, and $T=2X^2+3X-1$.
}
{} {}
\inputexercise
{}
{
Let $K$ be a field and let $K[X]$ be the polynomial ring over $K$. Show that every polynomial
\mathl{P \in K[X],\, P \neq 0,}{} can be decomposed as a product
\mathdisp {P= (X- \lambda_1)^{\mu_1} \cdots (X- \lambda_k)^{\mu_k} \cdot Q} { }
where
\mathl{\mu_j \geq 1}{} and $Q$ is a polynomial with no roots (no zeroes). Moreover the different numbers
\mathl{\lambda_1 , \ldots , \lambda_k}{} and the exponents
\mathl{\mu_1 , \ldots , \mu_k}{} are uniquely determined apart from the order.
}
{} {}
\inputexercise
{}
{
Let $F \in {\mathbb C}[X]$ be a \definitionsverweis {non-constant}{}{} \definitionsverweis {polynomial}{}{.} Prove that $F$ can be decomposed as a product of \definitionsverweis {linear factors}{}{.}
}
{} {}
\inputexercise
{}
{
Determine the smallest real number for which the \definitionsverweis {Bernoulli inequality}{}{} with exponent $n=3$ holds.
}
{} {}
\inputexercise
{}
{
Sketch the graph of the following \definitionsverweis {rational functions}{}{} \maabbdisp {f=g/h} {U} {\R } {,} where each time $U$ is the \definitionsverweis {complement set}{}{} of the set of the zeros of the denominator polynomial $h$. \aufzaehlungsieben{$1/x$, }{$1/x^2$, }{$1/(x^2+1)$, }{$x/(x^2+1)$, }{$x^2/(x^2+1)$, }{$x^3/(x^2+1)$, }{$(x-2)(x+2)(x+4)/(x-1)x(x+1)$. }
}
{} {}
\inputexercise
{}
{
Let $P\in \R[X]$ be a \definitionsverweis {polynomial}{}{} with \definitionsverweis {real}{}{} coefficients and let $z \in {\mathbb C}$ be a \definitionsverweis {root}{}{} of $P$. Show that also the \definitionsverweis {complex conjugate}{}{} $\overline{ z }$ is a root of $P$.
}
{} {}
\zwischenueberschrift{Hand-in-exercises}
\inputexercise
{3}
{
Consider the set $M=\{1,2,3,4,5,6,7,8\}$ and the function
\maabbeledisp {\varphi} {M} {M
} {x} {\varphi(x)
} {,} defined by the following table
\wertetabelleachtausteilzeilen { $x$ }
{\mazeileundfuenf {1} {2} {3} {4} {5} }
{\mazeileunddrei {6} {7} {8} }
{ $\varphi(x)$ }
{\mazeileundfuenf {2} {5} {6} {1} {4} }
{\mazeileunddrei {3} {7} {7} }
Compute $\varphi^{1003}$, that is the $1003$-rd composition
\zusatzklammer {or \stichwort {iteration} {}} {} {} of $\varphi$ with itself.
}
{} {}
\inputexercise
{2}
{
Prove that a strictly increasing function
\maabbdisp {f} {\R} {\R } {} is injective.
}
{} {}
\inputexercise
{3}
{
Let $L,M,N$ be sets and let
\mathdisp {f:L \longrightarrow M \text{ and } g:M \longrightarrow N} { }
be
\definitionsverweis {functions}{}{}
with their
\definitionsverweis {composite}{}{}
\maabbeledisp {g \circ f} {L} {N
} {x} {g(f(x))
} {.}
Show that if $\verknuepfung {f} {g}$ is
\definitionsverweis {surjective}{}{,}
then also $g$ is surjective.
}
{} {}
\inputexercise
{3}
{
Calculate in the
\definitionsverweis {polynomial ring}{}{} ${\mathbb C}[X]$ the product
\mathdisp {((4+i)X^3-iX^2+2X+3+2i) \cdot ( (2-i)X^3+(3-5i)X^2+(2+i)X+1+5i)} { . }
}
{} {}
\inputexercise
{3}
{
Perform, in the polynomial ring ${\mathbb C}[X]$ the division with remainder $\frac{P}{T}$, where $P= (5+i)X^4+iX^2+(3-2i)X-1$ and $T=X^2+iX+3-i$.
}
{} {}
\inputexercise
{5}
{
Let
\mathl{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
\mathkor {} {1} {or} {2} {.}
}
{} {}
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