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\newtheorem{cor}[thm]{Corollary}
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\newtheorem{exer}{Exercise}



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\title{Math 3100 Homework Assignment 1 - Due Wednesday February 3rd}

\author{Your name here}


\date{\today}

\begin{document}

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\begin{exer}
Give an uncountable family of infinite subsets of $\mathbb N$ such that any distinct two of these sets intersect in a set with no more than 2021 elements, or else prove that no such family exists. 
\end{exer}



\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}


\begin{exer}
Let $A$ be a set. The {\bf power set} of $A$ is the set $\mathcal P(A)$ that contains all subsets of $A$. For example $\mathcal P(\emptyset) = \{ \emptyset \}$, and $\mathcal P(\{ 0, 1 \}) = \{ \emptyset, \{ 0 \}, \{ 1 \}, \{ 0, 1 \} \}$. Prove that for any set $A$ there is no surjective function $f: A \to \mathcal P(A)$. 
\end{exer}




\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

\begin{exer}
A real number is called {\bf algebraic} if it is a root of a polynomial having integer coefficients. For example any rational number $\frac{a}{b}$ is a root of the polynomial $b x - a$, or the number $\sqrt{2} + 1$ is a root of the polynomial $(x - 1)^2 - 2 = x^2 -2x - 1$. Prove that there exist real numbers that are not algebraic. 
\end{exer}





\noindent\makebox[\linewidth]{\rule{\textwidth}{0.4pt}}

\begin{exer}
Prove that for every natural number $n \in \mathbb N$ the number $n^3 + 5n$ is divisible by $6$. 
\end{exer}





%\begin{thebibliography}{9}



%\bibitem{halmosmath}
%Paul R. Halmos
%How to write mathematics.
%\url{https://sites.math.washington.edu/~lind/Resources/Halmos.pdf}

%\bibitem{knuthmath}
%Donald E. Knuth, Tracy Larrabee and Paul M. Roberts
%Mathematical Writing.
%\url{https://jmlr.csail.mit.edu/reviewing-papers/knuth_mathematical_writing.pdf}

%\end{thebibliography}

\end{document}

