hw-subset.tex

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\probsec{~\ref{sec:subsets}}
\begin{enumerate}
  \item Let $R = \{\text{rectangles}\}$ and $S = \{\text{squares}\}$. Determine if $R \subset S$.

  \item Let $E = \{2n \mid n \in \Z\}$ and $P = \{n \in \Z \mid n \text{ is prime}\}$. Prove that $E \not\subset P$ and $P \not\subset E$.\sidenote{This is a two-part problem!}

  \item In each of the following, you are given two sets $A$ and $B$. Determine if $A \subset B$, $B \subset A$, both, or neither.
\begin{enumerate}
    \item $A = \{1, 2\}$, $B = \{2\}$
    \item $A = \emptyset$, $B = \R$
    \item $A = \{\{1,2\}, 3\}$, $B = \Z$
    \item $A = \{c \in \R | \exists a, b \in \R \text{ such that } a^2 - b^2 = c\}$, $B = \R$
\end{enumerate}


  \item Suppose $A$, $B$, and $C$ are sets with $A \subset B$ and $B \subset C$. Prove that $A \subset C$.

  \item Suppose $A$ and $B$ are finite sets with $A \subset B$. What can you say about $\# A$ and $\# B$? You do not have to prove your answer.

  \item A set $A$ has exactly one subset. What can you say about $A$?

    \item Determine which of the following statements is true.
  \begin{enumerate}
      \item $\exists n \in \N$ such that $\forall X \subset \N$, $n \geq \# X$.
      \item $\exists X \subset \N$ such that $\forall n \in \N$, $n \geq \# X$.
  \end{enumerate}
\end{enumerate}