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{\large \sf MATH 280 Discrete Mathematical Structures \hfil
Assignment \#1 } \hfill Name: \textsf{Rick Halterman}

{\small The point values for each question is given within [].
        The total number of points for this assignment is 65.}
\reversemarginpar
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\begin{enumerate}
%%  \setlength{\itemsep}{1.5cm}



    \item 
    Describe each of the following sets by listing its elements:
    \points{12}

    \begin{enumerate}

        \item $\{x ~|~ x \in \mathbb{N} \mbox{ and } x^2 - 5x + 6 = 0\}$

              [Your answer here.]

        \item $\{\cfrac{1}{x} ~|~ x \in \{ 2, 4, 6, 8\}\}$

              [Your answer here.]

        \item $\{x ~|~ x \in \mathbb{R} \mbox{ and } x^2 = 7\}$

              [Your answer here.]

        \item $\{x ~|~ x \in \mathbb{N} \mbox{ and } x^2 - 2x - 8 = 0\}$

              [Your answer here.]

        \item $\{x^2 ~|~ x \in \{-2, -1, 0, 1, 2\}\}$

              [Your answer here.]

        \item $\{x \in \mathbb{P} ~|~ x \mbox{ is a factor of 24 } \}$

              [Your answer here.]


    \end{enumerate}

    \item Let
          \points{42}
          \begin{align*}
             A &= \{2,4,5,6,8\} \\
             B &= \{1,4,5,9\} \\
             C &= \{x~|~x \in \mathbb{Z} \mbox{ and } 2 \le x < 5\}
          \end{align*}
          be subsets of $U = \{0,1,2,3,4,5,6,7,8,9\}$.  Find
          \begin{enumerate}
            \item $A \cup B$ 

              [Your answer here.]

            \item $A \cap B$ 

              [Your answer here.]

            \item $A \cap C$ 

              [Your answer here.]

            \item $B \cup C$ 

              [Your answer here.]

            \item $A - B$ 

              [Your answer here.]

            \item $A^{C}$ 

              [Your answer here.]

            \item $A \cap A^{C}$ 

              [Your answer here.]

            \item $(A \cap B)^{C}$ 

              [Your answer here.]

            \item $C - B$ 

              [Your answer here.]

            \item $(C \cap B) \cup A^{C}$ 

              [Your answer here.]

            \item $(B - A)^{C} \cap (A - B)$ 

              [Your answer here.]

            \item $(C^{C} \cap B)^{C}$ 

              [Your answer here.]

            \item $U^{C}$  

              [Your answer here.]

            \item $B \times C$  

              [Your answer here.]

            \item $\wp(B)$ 

              [Your answer here.]

            \item $A \oplus B$

              [Your answer here.]

            \item $B \oplus C$

              [Your answer here.]

            \item $|U|$

              [Your answer here.]

            \item $|U \times U|$

              [Your answer here.]

            \item $|\wp(U \times U)|$

              [Your answer here.]

            \item $|\varnothing|$

              [Your answer here.]

            \item $|\{\varnothing\}|$

              [Your answer here.]

          \end{enumerate}

    \item Let
          \points{8}
          \begin{align*}
            A &= \{ x ~|~ x 
                  \mbox{ is a word that appears before \textit{dog}
                   in an English language dictionary} \} \\
            B &= \{ x ~|~ x 
                  \mbox{ is a word that appears after \textit{canary}
                   in an English language dictionary} \} \\
            C &= \{ x ~|~ x \mbox{ is a word of more than 
                       four characters} \}  
          \end{align*}
         Which of the following are true statements?
          \begin{enumerate}
            \item $B \subseteq C$     [Your answer here.]
            \item $A \cup B = \{ x~|~ x \mbox{ is a word in an 
                  English language dictionary } \}$     
                  \hspace{0.2in} [Your answer here.]
            \item \textit{cat} $\in B \cap C^{C}$     
                  \hspace{0.2in} [Your answer here.]
            \item \textit{bamboo} $\in A - B$     
                  \hspace{0.2in} [Your answer here.]
          \end{enumerate}

    \item Suppose \points{6} 
    $U$  is an infinite universal set, and $A$ and
    $B$ are infinite subsets of $U$.  Answer the following
    questions and justify your answer.
    \begin{enumerate}
    \item Must $A^{C}$ be infinite?
                  \hspace{0.2in} [Your answer here.]

    \item Must $A \cup B$ be infinite?
                  \hspace{0.2in} [Your answer here.]

    \item Must $A \cap B$ be infinite?
                  \hspace{0.2in} [Your answer here.]
    \end{enumerate}

    \item Find \points{8} 
          the binary representation of each of the following 
          positive integers: 
          \begin{enumerate}
          \item 64 
                  \hspace{0.2in} [Your answer here.]
          \item 67 
                  \hspace{0.2in} [Your answer here.]
          \item 28 
                  \hspace{0.2in} [Your answer here.]
          \item 256 
                  \hspace{0.2in} [Your answer here.]
          \end{enumerate}


    \item Calculate \points{10}
    the following series:
    \begin{enumerate}
        \item $\sum\limits_{i=1}^{10} {i}$

                  [Your answer here.]

        \item $\sum\limits_{i=1}^{5} {20}$

                  [Your answer here.]

        \item $\sum\limits_{j=1}^{5} {(2j + 1)}$

                  [Your answer here.]

        \item $\sum\limits_{i=-n}^{n} {i}$, for $n=1, 2, 3, 4$

                  [Your answer here.]

        \item $\prod\limits_{i=1}^{5} {i}$

                  [Your answer here.]

    \end{enumerate}

    \item Draw \points{2}
    the Venn diagram for $\bigcap\limits_{i=1}^{4}{A_i}$
                  
                  [Your answer here.]



\end{enumerate}


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