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\begin{document}

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  \firstpageheader{ Math 15A }{ Homework 4 }{ February 9, 2019 }
  \runningheader{ Math 15A }{ Homework 4, Page \thepage\ of \numpages}{ February 9, 2019 }

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    \makebox[0.4\textwidth]{Name: \enspace\hrulefill}

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    \makebox[0.4\textwidth]{Pid:\enspace\hrulefill}
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  \begin{questions}
    \question[10]
      Let us define a union of more than two sets as follows.
			Let $A_1$, \dots, $A_n$ be some sets. Then
			\begin{itemize}
			  \item $\bigcup_{i = 1}^1 A_i = A_1$ and
			  \item $\bigcup_{i = 1}^{k + 1} A_i = \left( \bigcup_{i = 1}^k A_i \right)
			    \cup A_{k + 1}$.
			\end{itemize}
			
			Show that $\bigcup_{i = 1}^n [i] = [n]$ for all integers $n > 0$.

      \begin{solution}[\stretch{1}]
      \end{solution}
      \newpage
    \question[10]
      Let us define an intersection of more than two sets as follows.
			Let $A_1$, \dots, $A_n$ be some sets. Then
			\begin{itemize}
			  \item $\bigcap_{i = 1}^1 A_i = A_1$ and
			  \item $\bigcap_{i = 1}^{k + 1} A_i = \left( \bigcap_{i = 1}^k A_i \right)
			    \cap A_{k + 1}$.
			\end{itemize}
			
			Show that $\bigcap_{i = 1}^n \{x \in \mathbb{N} ~:~ i \le x \le n \} = \{n\}$
			for all integers $n > 0$.

      \begin{solution}[\stretch{1}]
      \end{solution}
      \newpage
  \end{questions}
\end{document}