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Mathematics-Online course: Prepcourse Mathematics - Basics - Exercises and Test

Exercises

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Check whether the following expressions are tautologies. Use transcriptions as well as truth tables.
(Authors: Hörner/Lesky/Abele)
The characters $ \diamondsuit,\,\square$ each symbolize one logical operator of the set $ \{\land,\lor,\Rightarrow\}$. In which cases does the following relation hold:

$\displaystyle A\,\diamondsuit\,(B\,\square\, C) \Leftrightarrow (A\,\diamondsuit\,B)\,\square\,(A\,\diamondsuit\,C) \ ?$

(Authors: Wipper/Abele)
Given are sets $ A,B\neq \emptyset$, a map $ f:A\rightarrow B$ as well as the subsets $ U,V\subset A$ and $ X,Y\subset B$. Consider the following relations; either prove them or find a suitable counter-example respectively.
a) $ f(U\cup V)=f(U)\cup f(V)$ b) $ f(U\cap V)=f(U)\cap f(V)$
c) $ f^{-1}(X\cup Y)=f^{-1}(X)\cup f^{-1}(Y)$                  d) $ f^{-1}(X\cap Y)=f^{-1}(X)\cap f^{-1}(Y)$
(Authors: Wipper/Abele)
Express the following statements about a map $ f: \mathbb{R}\longrightarrow\mathbb{R}$ in the formal way:

a) $ f$ is not surjective  b) $ f$ is not injective
c) $ f$ is not bijective  d) $ f$ is neither surjective nor injective
(Authors: Apprich/Höfert)
Use induction to proof for $ n\in\mathbb{N}$ :


a)      $ \displaystyle \sum_{k=1}^n k^3 = \frac{n^2(n+1)^2}{4} $           b)      $ \displaystyle \sum_{k=1}^n \frac{1}{k(k+1) }=\frac{n}{n+1}$

(Authors: Kimmerle/Roggenkamp/Höfert)
Prove via mathematical induction for $ n\in\mathbb{N}$:
a)
$ a_n=n^3 + 5n$ can be divided by $ 6$.
b)
$ b_n=\displaystyle{\frac{n}{3} + \frac{n^2}{2}+\frac{n^3}{6}}$ is a natural number.

(Authors: Boßle/Wipper/Abele)
How many possibilites do exist to express $ n\in\mathbb{N}$ as a sum of $ k$ natural numbers, if the order of summands is taken into account?
(Authors: Höllig/Abele)
Prove the following equations and illustrate the solutions in Pascal's Triangle.
a) $ {\displaystyle{\sum_{k=0}^n \left(\begin{array}{c} m+k\\ k \end{array}\right) = \left(\begin{array}{c} m+n+1\\ n \end{array}\right)}}$          b) $ {\displaystyle{\sum_{\ell=1}^n \left(\begin{array}{c} n+k-\ell \\ k \end{array}\right) = \left(\begin{array}{c} n+k\\ k+1 \end{array}\right)}}$
(Authors: Höllig/Abele)
How many possibilities do exist to be dealt a Full-House (three of a kind plus a pair) in poker with 32 cards (8 cards in 4 suits) ?
\includegraphics{fullhouse}
How many possibilities do exist to improve the displayed hand into four of a kind (four cards of the same denomination) by changing two cards?
(Authors: Höllig/Hörner/Abele)
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  automatically generated 10/23/2009