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\title{Problem Set 4}
\author[Daniel Gonzalez Cedre]{Discrete Mathematics}
\publisher{University of Notre Dame}
\date{Due on the \red{20\textsuperscript{th} of February, 2024}}

\begin{document}

\maketitle

\begin{enumerate}
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  \item[(20 pts) \quad 1.]
    \begin{enumerate}
      \item
        Show $\forall x \term*{\emptyset \subseteq x}$.
      \item
        Show $\forall x \term*{x \subseteq x}$.
      \item
        Show $\forall x \term*{\emptyset \in \power{x}}$.
      \item
        Show $\forall x \term*{x \in \power{x}}$.
      \item
        Show $\forall x \forall y \forall z \term*{\term*{(x \subseteq y) \land (y \subseteq z)} \implies x \subseteq z}$.
    \end{enumerate}
  \item[(10 pts) \quad 2.]
    We define the \defn{intersection} and \defn{difference} of any two sets $x$ and $y$ below.
    \begin{align*}
      x \intersect y &\defeq \set*{z \suchthat z \in x \land z \in y} \\
      x \setminus y &\defeq \set*{z \suchthat z \in x \land z \not \in y}
    \end{align*}
    \begin{enumerate}
      \item
        Show $\forall x \forall y \exists z \term*{z = x \intersect y}$.
      \item
        Show $\forall x \forall y \exists z \term*{z = x \setminus y}$.
    \end{enumerate}
  \item[(20 pts) \quad 3.]
    We define the \defn{union} of any two sets $x$ and $y$ below.
    \begin{equation*}
      x \union y \defeq \set*{z \suchthat z \in x \lor z \in y}
    \end{equation*}
    \begin{enumerate}
      \item
        Show $\forall x \forall y \term{x \intersect y \subseteq x}$.
      \item
        Show $\forall x \forall y \term{x \subseteq x \union y}$.
      \item
        Show $\forall x \forall y \term*{\power{x} \union \power{y} \subseteq \power{x \union y}}$.
      \item
        Show $\forall x \forall y \term*{x \intersect y = x \iff x \in \power{y}}$.
    \end{enumerate}
  \item[(50 pts) \quad 4.]
    We define the \defn{union over $x$} and \defn{intersection over $x$} for any set $x$ below.
    \begin{align*}
      \union x &\defeq \set*{z \suchthat \exists y \term*{y \in x \land z \in y}} \\
      \intersection x &\defeq \set*{z \suchthat \forall y \term*{y \in x \implies z \in y}}
    \end{align*}
    \begin{enumerate}
      \item
        Show that $\forall x \term*{\union \power{x} = x}$.
      \item
        What is $\union \emptyset$?
        Justify your answer with a proof.
      \item
        What is $\intersection \emptyset$?
        Justify your answer with a proof.
      \item
        Is $\emptyset = \set{z \suchthat z \in \emptyset}$?
        Justify your answer with a proof.
      \item
        Is $\emptyset = \set{z \suchthat z \not \in \emptyset}$?
        Justify your answer with a proof.
    \end{enumerate}
\end{enumerate}

\end{document}
