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\begin{document}

\noindent
{\Large 
Sets Worksheet  \hfill Name:\underline{\hspace{2.8in}}\\ \\
MA~201}
\vspace{.25in}

\noindent
{\bf WARNING:}  You must {\bf SHOW ALL OF YOUR WORK}.  You will receive 
NO CREDIT if you do not show your work.

\be 
\item Let $U= \set{0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}$ be the universe.
Let $A=\set{x \in U |\, x \text{ is even}}$, $B = \set{x \in U|\, 1 \leq x <
10}$, and $C = \set {x \in U |\, x \text{ is even or } x = 15}$.  Find the
following.
\be
	\item $n(A)$
	\item $n(B)$
	\item $n(A \cup B)$
	\item $n(A \cap B)$  \\
	(What relationship do you notice between
	$n(A)$, $n(B)$, $n(A\cup B)$, and $n(A \cap B)$?  Does this
	relationship always hold? If so, explain why it always holds.  If
	not, provide a counterexample.)
	\item $A \cap B$
	\item $A \cup \ol{C}$
	\item $\ol{A \cup C}$
	\item $(A \cup \ol{B}) \cap C$
	\ee
\item True or False.
	\be
	\item \spc $\set{1,2} = \set{2,1}$
	\item \spc $\set{1,2} \sim \set{2,1}$
	\item \spc $\set{1,2} \sim \set{3,4}$
	\item \spc $(1,2) = (2,1)$
	\item \spc Let $B = \set {a, b, c, d, e}$.  Then $B = 5$.
	\item \spc $\emptyset \subseteq \set {a,b,c}$
	\item \spc $0 \div 21$ is defined.
	\item \spc $21 \div 0$ is defined.
	\item \spc $\set{a,b,c} \cup \emptyset = \set{a,b,c}$
	\item \spc $\set{a,b,c} \cap \emptyset = \set{a,b,c}$
	\item \spc $\emptyset = \set{0}$
	\item \spc $n(\emptyset) = 0$
	\item \spc 
		\begin{minipage}[t]{5in}
		Let $A= \set{x |\, x \text{ is an even whole number}}$. \\
		Let $B = \set{y |\, y \text{ is an even natural number}}$.  
		Then $B \subseteq A$.
		\end{minipage}
	\item \spc 
		\begin{minipage}[t]{5in}
		Let $A= \set{x |\, x \text{ is an even whole number}}$. \\
		Let $B = \set{y |\, y \text{ is an even natural number}}$.  
		Then $B \subset A$.
		\end{minipage}
	\ee
\item True or False.  If the statement is true, briefly explain why it is
true.  If it is false, provide a counterexample.
	\be
	\item \spc  If $A$ and $B$ are finite sets, then $n(A) + n(B) = n(A
		\cup B)$.
	\item \spc  If $A$ and $B$ are finite sets, then $n(A) \times n(B)
		= n(A \times B)$
	\item \spc If $n(A \cap B) < n(A)$, then $B \subset A$.
	\item \spc If $A \subseteq B$ and $B \subset C$, then $A \subset
		C$.
	\ee
\item 
	\be
	\item Show that the set of whole numbers, $W$, is equivalent to the 
		set of natural numbers, $N$, by carefully describing a 
		one-to-one correspondence between the sets.
	\item According to the one-to-one correspondence you described in
		part (a), which whole number is paired with the natural
		number $999$?
	\item According to the one-to-one correspondence you described in
		part (a), which natural number is paired with the whole
		number $999$?
	\item According to the one-to-one correspondence you described in
		part (a), which whole number is paired with the natural
		number $x$?
	\item According to the one-to-one correspondence you described in
		part (a), which natural number is paired with the whole
		number $y$?
	\ee
\item 
	\be
	\item Show that the set of even whole numbers, $E$, is equivalent to 
		the set of odd whole numbers, $O$, by carefully describing a 
		one-to-one correspondence between the sets.
	\item According to the one-to-one correspondence you described in
		part (a), which even number is paired with the odd 
		number $999$?
	\item According to the one-to-one correspondence you described in
		part (a), which odd number is paired with the even 
		number $764$?
	\item According to the one-to-one correspondence you described in
		part (a), which even number is paired with the odd 
		number $m$?
	\item According to the one-to-one correspondence you described in
		part (a), which odd number is paired with the even 
		number $n$?
	\ee
\item Do number 11 on page 94 of your textbook.
\item Let $A = \set {a,b,c}$ and $B = \set{c,d,ef}$.  Then $n(A) = 3$,
$n(B) = 4$, and $n(A \cup B) = 6$ (Why?)  Look at the definition for
addition of whole numbers given on page 99 of your textbook.  In this
example, $n(A) + n(B) \neq n(A \cup B)$.  Is there a conflict between this
example and the definition for addition of whole numbers?  Why or why not?
Explain briefly.
\item Use the measurement model to illustrate the following.
	\be
	\item $4+6 = 6+4$
	\item $4 \times (2+3) = 4 \times 2 + 4 \times 3$
	\ee
\item True or False.  Briefly justify your answer.
	\be
	\item \spc $\set{0,1}$ is closed under multiplication.
	\item \spc $\set{0,1}$ is closed under subtraction.
	\item \spc $\set{0,1}$ is closed under addition.
	\item \spc Let $X$ be s subset of the whole numbers that contains
	$2$ and $4$.  If $X$ is closed under addition then $3$ cannot be
	an element of $X$.
	\ee
\item Let $X$ be a subset of the whole numbers that contains $2$.  If $X$
is closed under addition, what whole numbers must be contained in $X$?
What, if any, numbers are certainly not contained in $X$?
\item Use the number line to illustrate the following facts.
	\be
	\item $15-2 = 13$
	\item $4*5 = 20$
	\ee
\item Do number 16 on page 109 of your textbook.
\item Do number 28 on page 111 of your textbook.
\item State the Division Algorithm.
\item Use sets to show that $6 < 9$.
\item Do units need to be the same when adding?  when subtracting? when
multiplying?  when dividing?
\item For each subtraction model, write a separate word problem that 
	illustrates $$53 - 26.$$
	\be
	\item The comparison model for subtraction
	\item The missing addend model for subtraction
	\item The take-away model for subtraction
	\item The measurement model for subtraction
	\ee
\item For each addition model, write a separate word problem that 
	illustrates $$53 + 26.$$
	\be
	\item The set model for addition
	\item The measurement model for addition
	\ee
\item For each multiplication model, write a separate word problem that 
	illustrates $$4 \times 3.$$
	\be
	\item The repeated addition model for multiplication
	\item The array model for multiplication
	\item The rectangular area model for multiplication
	\item The multiplication tree model for multiplication
	\item The Cartesian product model for multiplication
	\ee
\item For each division model, write a separate word problem that 
	illustrates $$35 \div 7.$$
	\be
	\item The repeated subtraction model for division
	\item The partition model for division
	\ee
\item Clearly explain why division by zero is undefined.
\item Clearly explain why $\frac {0}{253}$ is defined.
\item Rewrite as a single whole number exponential, if possible.
	\be
	\item $\bfrac{7^{15}}{7^4}$
	\item $2^5 - 2^3$
	\item ${(2^5)}^6$
	\item $4^5 \times 6^5$
	\ee
\item Evaluate $3^0$.
\item Do number 25 on page 131 of your textbook.
\item Let $X = \set{1,2,3}$ and $Y = \set{a,b}$. 
	\be
	\item Find $X \times Y$.
	\item Find $Y \times X$.
	\item Is $X \times Y = Y \times X$?
	\item Is $n(X \times Y) = n(Y \times X)$?
	\ee
\item Let $X = \set{a,b}$ and $Y = \set{b,a}$. 
	\be
	\item Find $X \times Y$.
	\item Find $Y \times X$.
	\item Is $X \times Y = Y \times X$?
	\item Is $n(X \times Y) = n(Y \times X)$?
	\ee
\item Let $A$ and $B$ be finite sets.
	\be
	\item What can be said about $n(A \times B)$ and $n(B \times A)$?
		Relate this observation to a property of whole number
		multiplication.
	\item If $A \times B = B \times A$, what must be true about $A$ and
	$B$?
	\ee
 
\ee

\end {document}
