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%%\swapnumbers
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\begin{document}
\allowdisplaybreaks


\raggedright
\section*{Sets and relations}



\begin{zad}
Let $A = \{1, 2, 5, 6, 7, 10, 11, 12\}$, $B = \{x \mid x \text{ is an odd integer and } 3 \leq x \leq 10\}$, and $C = \{a, b, c, 4, 5, 6\}$.
\begin{enumerate}[label=\rm(\alph*)]
\item Fill in the blanks $\,\_\!\_\!\_\,$ with the most suitable symbol $(\in, \notin, \subseteq, \not\subseteq)$:\newline
$7\,\_\!\_\!\_\,A$, $6\,\_\!\_\!\_\,B$, $10\,\_\!\_\!\_\,B$, $4\,\_\!\_\!\_\,A$, $b\,\_\!\_\!\_\,C$, $b\,\_\!\_\!\_\,B$, $\{2, 5, 12\}\,\_\!\_\!\_\,A$, $\{3, 5, 6\}\,\_\!\_\!\_\,B$, $\{a, b, 5\}\,\_\!\_\!\_\,C$.
\item Compute $A \cup B$, $A \setminus B$, $\mathcal{P}(A \cap B)$, $(A \cap B) \times \{a, b\}$.
\end{enumerate}
\end{zad}

\begin{zad}
\begin{enumerate}[label=\rm(\alph*)]
\item Let $A$ and $B$ be given sets. Show that $A \cap (B \setminus A) = \emptyset$. Explain each step of your proof.
\item Let $A$ and $B$ be subsets of the universal set $U$. Show that $A \subseteq B$ if and only if $\overline{B} \subseteq \overline{A}$. Explain each step of your proof.
\end{enumerate}
\end{zad}

\begin{zad}
Justify whether the statement 
$$
(A \cup B) \cap (A \cup C) \subseteq \overline{A} \cup C
$$
holds for arbitrary sets $A$, $B$, and $C$. Explain each step of your proof.
\end{zad}


\begin{zad}
Let $A$ be a non-empty set. Determine which of the sets 
$$
\emptyset, \{\emptyset\}, A, \{A\}, \{A, \emptyset\}
$$ 
are elements and which are subsets of the sets $\mathcal{P}(A)$ and $\mathcal{P}(\mathcal{P}(A))$.
\end{zad}

\begin{zad}
Justify whether the following statement is true or false:

\begin{enumerate}[label=\rm~]
\item If $\P(X) = \P(Y)$ for sets $X$ and $Y$, then $X$ and $Y$ are equal.
\end{enumerate}
\end{zad}

\begin{zad}
In the power set $\P(M)$ of the set $M = \{1, 2\}$, we introduce a relation $R$ defined by the rule:
$$
ARB \qquad \Longleftrightarrow \qquad
A \cup \{1\} = B \cup \{2\}.
$$
Write all ordered pairs in $R$.
\end{zad}

\begin{zad}
On the set of real numbers $\RR$, we define a relation $\sim$:
$$
a \sim b \qquad \Longleftrightarrow \qquad a - b \in \ZZ.
$$
Justify whether $\sim$ is an equivalence relation. If $\sim$ is an equivalence relation, find the equivalence classes $[\frac{1}{2}]_\sim$ and $[\sqrt{2}]_\sim$.
\end{zad}




\section*{Mathematical Induction}

\begin{zad}
Show that the number $3^{7^{999}} - 1$ is divisible by $2$. Solve the problem using mathematical induction.
\end{zad}

\begin{zad}
Prove that for every natural number $n > 1$, the equality
$$
\left(1 - \frac{1}{4}\right) \cdot \left(1 - \frac{1}{9}\right) \cdot \left(1 - \frac{1}{16}\right) \cdot \ldots \cdot \left(1 - \frac{1}{n^2}\right) = \frac{n + 1}{2n}
$$
holds.
\end{zad}

\begin{zad}
Show that the sum $1 + 3 + 5 + 7 + \cdots + (2 \cdot 7^7 - 1)$ is divisible by $7^7$. Solve the problem using mathematical induction.
\end{zad}

\begin{zad}
Prove that the number $37^{500} - 37^{100}$ is divisible by $10$.
\end{zad}


\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}




\newpage
\section*{Functions}

\begin{zad}
Assume that the set $A$ has exactly two elements and the set $B$ has exactly three elements. 
\begin{enumerate}[label=\rm(\alph*)]
\item Provide an example of a function $f: A \to B$. Define the inverse function. For the given example, determine if the inverse function $f^{-1}: B \to A$ exists, and if it does, find it. If it does not exist, explain why.
\item Provide an example of a function $g: B \to A$. For the given example, determine if the inverse function $g^{-1}: A \to B$ exists, and if it does, find it. If it does not exist, explain why.
\item How many functions map $A$ to $B$? How many of these functions are surjective? How many are injective? How many are bijective?
\end{enumerate}
\end{zad}

\begin{zad}
Let $\RR$ be the set of real numbers, and let $f: \RR \to \RR$ be a function defined by the rule
$$
f(x) = 5x + 12.
$$
Analyze the injectivity, surjectivity, and determine the inverse function $f^{-1}(x)$.
\end{zad}

\begin{zad}
Let $f : \ZZ \to \ZZ$ be a function defined by the rule
$$
f(x) = x^2.
$$
Analyze the injectivity, surjectivity, and bijectivity of the function $f$.
\end{zad}

\begin{zad}
Find an example of:
\begin{enumerate}[label=\rm(\alph*)]
\item An injective function $f: \NN \to \NN$ that is not surjective.
\item A function $f: \NN \to \NN$ that is surjective but not injective.
\end{enumerate}
\end{zad}




\section*{Absolute Value}

\begin{zad}
Solve the inequality
$$
\frac{|x+1|}{-x^2 - 2x + 15} > 0.
$$
\end{zad}

\begin{zad}
Solve the inequality
$$
x^2 - 14 < |x+2|.
$$
\end{zad}

\begin{zad}
Solve the inequality
$$
\left| \frac{x+2}{x-1} \right| > 2.
$$
\end{zad}

\begin{zad}
Solve the inequality in the set of real numbers $\RR$:
$$
\sqrt{x^2 - 1} \leq \sqrt{|x - 1|}.
$$
Write the solution in terms of intervals.
\end{zad}

\begin{zad}
Solve the inequality in the set of real numbers $\RR$:
$$
\sqrt{x^2 - 2x - 1} \leq \sqrt{|x - 1|}.
$$
Write the solution in terms of intervals.
\end{zad}




\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}






\newpage
\section*{Composition of Functions}

\begin{zad}
Two functions are given:
$$
f(x) = \frac{x}{\sin x}
\qquad \text{and} \qquad
g(x) = x^2 - 2x + \frac{1}{x^2} - \frac{2}{x}.
$$
\begin{enumerate}[label=\rm(\roman*)]
\item Determine the maximal (natural) domain of definition for both functions. 
\item Show that $g\left(\frac{1}{f(x)}\right) = (g \circ f)(x)$. Provide a detailed justification.
\end{enumerate}
\end{zad}

\begin{zad}
Given the functions $f(x) = \frac{1}{x^2 + 1}$ and $g(x) = \sqrt{x}$, compute the formulas and domains of the compositions $f \circ g$ and $g \circ f$.
\end{zad}

\begin{zad}
Let $f$ and $g$ be real functions of a real variable, defined as follows:
$$
f(x) = \begin{cases} 
-1, & x < 1, \\ 
1, & x \geq 1,
\end{cases}
\qquad
g(x) = \begin{cases} 
x^2, & x < 0, \\ 
x^2 - x - 1, & x \geq 0.
\end{cases}
$$
Determine the composition $f \circ g$.
\end{zad}

\begin{zad}
Let $f$ and $g$ be real functions of a real variable, defined as follows:
$$
f(x) = \begin{cases} 
-x^2 + 1, & x \geq 0, \\ 
e^x, & x < 0,
\end{cases}
\qquad
g(x) = \begin{cases} 
1, & x < 1, \\ 
-1, & x \geq 1.
\end{cases}
$$
Determine the compositions $g \circ f$ and $f \circ g$.
\end{zad}

\begin{zad}
Let $f$ and $g$ be real functions of a real variable, defined as follows:
$$
f(x) = \begin{cases} 
x, & x < 0, \\ 
0, & x \geq 0,
\end{cases}
\qquad
g(x) = \begin{cases} 
1, & |x| \geq \frac{\pi}{2}, \\ 
|\sin x|, & |x| < \frac{\pi}{2}.
\end{cases}
$$
Write the rule for the composition $g \circ f$.
\end{zad}






\section*{Rational Numbers}

\begin{zad}
Prove that the number $\sqrt{5}$ is irrational.
\end{zad}

\begin{zad}
Prove that the number $\sqrt{7}$ is irrational.
\end{zad}



\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}



\newpage

\section*{Real Numbers}

\begin{zad}
For the set $A$ in the set of real numbers, determine the supremum, infimum, minimum, and maximum, if they exist:
$$
A = \left\{1 - \frac{1}{n} \mid n \in \NN \right\}.
$$
Also, provide examples of a lower bound and an upper bound. Provide a detailed explanation of why the specified value is the infimum or supremum, or explain why the set does not have an infimum or supremum.
\end{zad}

\begin{zad}
For the set $S$ in the set of real numbers, determine the supremum, infimum, minimum, and maximum, if they exist:
$$
S = \left\{x \in \RR : |x| + 2 \geq 2x \right\}.
$$
Also, provide examples of a lower bound and an upper bound. Provide a detailed explanation of why the specified value is the infimum or supremum, or explain why the set does not have an infimum or supremum.
\end{zad}

\begin{zad}
For the sets $A$ and $B$ in the set of real numbers, determine the supremum, infimum, minimum, and maximum, if they exist:
\begin{enumerate}[label=\rm(\alph*)]
\item $A = \{x \mid x^2 \leq 11\}$.
\item $B = \{x \in [2,5] \mid x \text{ contains at least two threes in its decimal representation}\}$.
\end{enumerate}
Also, provide examples of a lower bound and an upper bound for $A$ and $B$. For part (b), give the definition of infimum and supremum and provide a detailed explanation of why the specified value is the infimum or supremum, or explain why the set does not have an infimum or supremum.
\end{zad}






\section*{Number Systems}

\begin{zad}
\begin{enumerate}[label=\rm(\alph*)]
\item Convert the decimal number $751$ into a base-$6$ number.
\item Convert the base-$6$ number $451$ into a decimal number.
\end{enumerate}
\end{zad}

\begin{zad}
\begin{enumerate}[label=\rm(\alph*)]
\item Convert the decimal number $588$ into a base-$6$ number.
\item Convert the base-$6$ number $544$ into a decimal number.
\end{enumerate}
\end{zad}




\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}




\newpage
\section*{Complex Numbers}

\begin{zad}
Let $z = 1 + i$.
\begin{enumerate}[label=\rm(\alph*)]
\item Express the complex number $z^{50}$ in polar form, and calculate its real and imaginary components.
\item Express the complex number $z^{-1}$ in polar form, and calculate its real and imaginary components.
\item The complex number $z = 1 + i$ can be represented by the point $T(1, 1)$ in the complex plane. Explain the geometric meaning of the product $z \cdot \left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right)$ with respect to the point $T$. What is the geometric meaning of the product $z \cdot \left(\cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4}\right)$ with respect to the point $T$?
\end{enumerate}
\end{zad}

\begin{zad}
Solve the equation in the set of complex numbers:
$$
z^2 = 2i.
$$
Write all solutions in the form $a + ib$, where $a$ and $b$ are real numbers, and also in polar form. Additionally, plot all solutions in the complex plane with rectangular coordinates.
\end{zad}

\begin{zad}
Find the solutions to the equation in the set of complex numbers $\CC$:
$$
2z^2 + 1 = 2i \, \text{\rm Im}(z).
$$
Write the solutions in polar form.
\end{zad}




\begin{zad}
Solve the equation in the set of complex numbers:
$$
z^3 = -4.
$$
Write all solutions in the form $a + ib$, where $a$ and $b$ are real numbers, and also in polar form. Additionally, plot all solutions in the complex plane using rectangular coordinates.
\end{zad}

\begin{zad}
Find all complex numbers $z$ that satisfy the equation $z^8 = 1$. Write all solutions in polar form $r(\cos\varphi + i\sin\varphi)$, where $r \in \RR$, $\varphi \in [0, 2\pi)$, and in rectangular form $x + iy$, where $x, y \in \RR$. Additionally, plot all solutions in the complex plane using rectangular coordinates.
\end{zad}



\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}



\newpage
\section*{Sequences}

\begin{zad}
Prove that the sequence with general term $\ds a_n = \frac{1}{\frac{n}{2n+1}}$ is:
\begin{enumerate}[label=\rm(\alph*)]
\item Monotonic,
\item Bounded above,
\item Bounded below,
\item Bounded,
\item Convergent.
\end{enumerate}
\end{zad}

\begin{zad}
A real sequence is given with the general term:
$$
a_n = 2 + \frac{1}{\frac{n}{n+1}}.
$$
Write the first, fifth, tenth, hundredth, and thousandth terms of the sequence. Justify whether the sequence is:
\begin{enumerate}[label=\rm(\alph*)]
\item Monotonic,
\item Bounded above,
\item Bounded below,
\item Bounded.
\end{enumerate}
If the sequence is convergent, find its limit.
\end{zad}

\begin{zad}
A real sequence is defined by the first term $x_1 = 1$ and the recursive relation:
$$
x_{n+1} = \frac{2x_n + 3}{x_n + 2}.
$$
Justify whether the sequence is:
\begin{enumerate}[label=\rm(\alph*)]
\item Monotonic,
\item Bounded above,
\item Bounded below,
\item Bounded.
\end{enumerate}
If the sequence is convergent, find its limit.
\end{zad}



\begin{zad}
Calculate the limit:
$$
\lim_{n\to\infty}(\sqrt{n-3}-\sqrt{n+3}).
$$
\end{zad}

\begin{zad}
A real sequence $(a_n)_{n\in{\mathbb N}}$ is given by the general term:
$$
a_n=\frac{n}{\sqrt{2+3n^2}}.
$$
\begin{enumerate}[label=\rm(\alph*)]
\item Show that the sequence $(a_n)_{n\in{\mathbb N}}$ is monotonic.
\item Show that the sequence $(a_n)_{n\in{\mathbb N}}$ is bounded.
\item Find the limit of the sequence $(a_n)_{n\in{\mathbb N}}$.
\end{enumerate}
\end{zad}

\begin{zad}
A real sequence $(a_n)_{n\in{\mathbb N}}$ is defined recursively by:
$$
a_{n+1}=5\sqrt{a_n-1}-3,
$$
with the first term $a_1 = 5$.
\begin{enumerate}[label=\rm(\alph*)]
\item Show that the sequence $(a_n)_{n\in{\mathbb N}}$ is monotonic.
\item Show that the sequence $(a_n)_{n\in{\mathbb N}}$ is bounded.
\item Find the limit of the sequence $(a_n)_{n\in{\mathbb N}}$.
\end{enumerate}
\end{zad}


\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}



\newpage
\section*{Function Limits}

\begin{zad}
Calculate the limit:
$$
\lim_{x\to 1} \frac{x^2 + x - 2}{- 2x^2 + 6x - 4},
$$
without using L'Hôpital's rule (which we studied in Analysis II).
\end{zad}

\begin{zad}
Calculate the limit:
$$
\lim_{x\to+\infty} x\,\arcctg x,
$$
without using L'Hôpital's rule (which we studied/will study in Analysis II).
\end{zad}


\begin{zad}
Calculate the limit 
$$
\ds{\lim_{\alpha\to\frac{\pi}{2}-0} \left(\sqrt{\tg\!^2\alpha + \frac{1}{\cos\alpha}} -\tg\alpha\right)}
$$
of course, calculate it {\bf without} using L'Hôpital's rule (which we have covered/will cover in Analysis II).
\end{zad}



\section*{Series}

\begin{zad}
Using partial fraction decomposition, sum the series
$$
\sum_{n=1}^{\infty} \frac{1}{(n+7)(n+8)}.
$$
\end{zad}

\begin{zad}
Determine precisely for which real numbers $a\in\RR$ the series
$$
\sum_{n=1}^{+\infty} \frac{a\,n^5}{2^n}
$$
converges (if such a number $a$ exists).
\end{zad}

\begin{zad}
Determine precisely for which real numbers $\alpha\in\RR$ the series
$$
\sum_{n=1}^{+\infty} \frac{\alpha}{\sqrt{n}}
$$
converges (if such a number $\alpha$ exists).
\end{zad}

\begin{zad}
Determine whether the given series converges or not
$$
\ds\sum_{n=1}^\infty \frac{1}{n(\sqrt{7}-2)^n}.
$$
\end{zad}

\begin{zad}
Determine whether the given series converges or not
$$
\ds\sum_{n=1}^\infty n! \left(\frac{2}{n}\right)^n.
$$
\end{zad}




\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}







\newpage
\section*{Continuity of Functions}

\begin{zad}
Determine the constant $k$ so that the function
$$
f(x)=
\left\{
\begin{array}{lcl}
5&;&x\in(-\infty,2),\\
x^2-10x+k&;&x\in[2,6],\\
-3&;&x\in(6,+\infty)\\
\end{array}
\right.
$$
is continuous on $\RR$, if possible.
\end{zad}

\begin{zad}
Determine the constants $\alpha$ and $\beta$ so that the functions 
$$
f(x)=\left\{\begin{array}{cc}
\arcctg\left(\frac{1}{x}\right), & x\ne 0\\
\alpha, & x=0
\end{array}\right.\qquad\mbox{and}\qquad
g(x)=\left\{\begin{array}{cc}
\arcctg(x), & x\ne 0\\
\beta, & x=0
\end{array}\right.
$$
are continuous on $\RR$, if possible (if such constants exist).
\end{zad}


\bigskip
\begin{center}
All above math problems  are taken from the following website: \url{https://osebje.famnit.upr.si/~penjic/teaching.html}.

{\sc The reader can find all solutions to the given problems on the same page.}
\end{center}


\end{document}
