set_theory_practice.tex

\yourname

\activitytitle{Set theory practice}{More practice working with sets}

\overview{Write really detailed proofs with crystal clear logic.  In particular, when showing that $A \subseteq B$, start with ``Let $x \in A$,'' show that $x$ is in $B$, and then say, ``Since $x \in A$ was arbitrary, $A \subseteq B$.''}

\show{Let $A, B$, and $C$ be sets.
Suppose that $A \subseteq B$ and $B \subseteq C$.
Show that $A \subseteq C$.
{\bf Note:} This {\em shows} transitivity but it does not {\em use} transitivity.}{1in}

\show{Let $A, B$, and $C$ be sets.
Suppose that $A \subset B$ and $B \subset C$.
Show that $A \subset C$.
{\bf Note:} Here we have strict set inclusion, so you will need to show that $A$ is not equal to $C$.
}{1in}

\show{Let $A, B$, and $C$ be sets.
Show that $C \subseteq A$ and $C \subseteq B$ if and only if $C \subseteq A \cap B$.
{\bf Note:}  ``If and only if'' means there are two things to show:
\blist{1in}
\item Suppose that $C \subseteq A$ and $C \subseteq B$.  Show that $C \subseteq A \cap B$.
\item Suppose that $C \subseteq A \cap B$.  Show that $C \subseteq A$ and $C \subseteq B$.
\elist
}{1in}

\show{Let $A$ and $B$ be sets.
Show that $A \cap B = B$ if and only if $B \subseteq A$.
}{2in}

\show{Let $A_1, A_2, A_3, \ldots$ and $B_1, B_2, B_3, \ldots$ be sets.
Suppose that $A_n \subseteq B_n$ for all $n = 1, 2, 3, \ldots$.
Show that $\bigcap_{n=1}^{\infty} A_n \subseteq \bigcap_{n=1}^{\infty} B_n$}{2in}

\show{Let $A_1, A_2, A_3, \ldots$ and $B_1, B_2, B_3, \ldots$ be sets.
Suppose that $A_n \subseteq B_n$ for all $n = 1, 2, 3, \ldots$.
Show that $\bigcup_{n=1}^{\infty} A_n \subseteq \bigcup_{n=1}^{\infty} B_n$}{2in}

\show{Let $A$ and $B_1, B_2, B_3, \ldots$ be sets.
Suppose that $A \subseteq \bigcap_{n=1}^{\infty} B_n$.  
Show that $A \subseteq B_n$ for all $n = 1, 2, 3, \ldots$.
Start the proof with ``Let $n \geq 1$ be an integer'' and be sure to end the proof by generalizing over $n$.
The second step in the proof is ``Let $x \in A$.''}{0in}

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