power_set_cartesian_product.tex

\yourname

\activitytitle{The power set and the Cartesian product}{Useful constructions with sets.}

\overview{The power set is our first example of thinking hard about collections of sets.  The Cartesian product is used often when you want ordered pairs or ordered triples of numbers or other objects.}

\problem{Write out the members of the following power sets.  It may be helpful to do \#3, \#4, then \#2, \#1, and finally \#5.
\blist{0.1in}
\item $S = \emptyset$.  ${\cal P}(S) = $
\item $S = \{ 1 \}$.  ${\cal P}(S) = $
\item $S = \{ 1,2 \}$.  ${\cal P}(S) = $
\item $S = \{ 1,2,3 \}$.  ${\cal P}(S) = $
\item $S = \{ 1,2,3,4 \}$.  ${\cal P}(S) = $
\item $S = \{ 1,2,3,4,5 \}$.  ${\cal P}(S) = $
\elist
}{0in}

\question{If $S$ has $n$ elements, how many members will ${\cal P}(S)$ have?  Explain as well as you can.}{1in}

\problem{Write the appropriate symbol between the entities, or mark the statement as true or false.  Give an explanation for anything that is not obvious enough.
\blist{0.0in}
\item $ 1 \quad\quad {\cal P}(\{ 1, 2, 3 \})$
\item $[3, 10] \quad\quad \Z$
\item $[3, 10] \quad\quad \R$
\item $\Q \quad\quad \R$
\item $\Q \quad\quad {\cal P}(\R)$
\item $[3, 10] \quad\quad {\cal P}(\R)$
\item $\N \quad\quad \R$
\item $\emptyset \quad\quad \R$
\item $\emptyset \quad\quad {\cal P}(\R)$
\item $\{ \emptyset \} \subseteq A$?
\item $\emptyset \subset {\cal P}(A)$?
\elist
}{0.0in}

\question{Suppose that $S$ is a set.  Then ${\cal P}(S)$ is also a set, but if we let $A \in {\cal P}(S)$, then $A$ is also a set.
Explain how this can be.
What is the relationship between $A$ and $S$?}{1.5in}

\problem{Let $I$ be a set, and for each $i$ in $I$, let $B_i$ be a set.
Show that ${\cal P}\left( \bigcap_{i \in I} B_i \right) = \bigcap_{i \in I} {\cal P}(B_i)$.\\
Let $A$ be an element of the set on the left-hand side.  Notice that $A$ is a set.  Argue that it is an element of the set on the right-hand side.
\vspace{1in}
Let $A$ be an element of the set on the right-hand side \ldots}{1in}

\problem{
\blist{0.75in}
\item Sketch the Cartesian product $A = [1,3] \times [2,5]$.
\item Sketch the Cartesian product $B = [2,4] \times [1,3]$.
\item Sketch the intersection $A \cap B$.
\item It seems that $A \cap B$ is also a Cartesian product.  Identify the sets whose product is $A \cap B$.
\item What is $([1,3] \cap [2,4]) \times ([2,5]\cap[1,3])$?
\elist
}{0.0in}

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