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{\bf Name: }
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{\bf Section: }

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\centerline {\bf Computer Science 243}
\centerline {\bf Spring 2025}
\centerline {\bf Homework 4}
% \centerline{\bf My name}
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\bf Due: 11:59 p.m., Wednesday, 2/19/25 \hfill Points: 50
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Answer the following questions and show your work.  Your final submission must be completely your own work.

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\item {[6 points]}  Use a direct proof to show that the product of an even and an odd integer is even. \\

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\item {[6 points]}  Use a proof by contraposition to show that if 3n + 7 is even, then n is odd. \\

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\item {[6 points]}  Use a proof method to prove or disprove that no odd integers are cubes.  State your proof method and show your proof. \\

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\item {[3 points each]}  Based on a domain of all letters in the English alphabet, let \\ \\
\hspace*{1cm}$A = \{s, c, i, e, n\}$ \\
\hspace*{1cm}$B = \{a, e, i, o, u\}$ \\
\hspace*{1cm}$C = \{c, o, m, p, u, t, e, r\}$ 
\begin{enumerate}[label=\alph*.]
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\item  Draw a Venn diagram showing the sets above. 
\item What is $\overline{A} \cap (C - B)$? 
\item What is $\mid \overline {A \cup B \cup C} \mid$? 
\item What is $P(B - C)$?
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\item {[3 points each]}  Based on the following sets, \\ \\
\hspace*{1cm}$A = \{\emptyset \}$ \\
\hspace*{1cm}$B = \{\emptyset, \{ \emptyset \} \}$ \\
\hspace*{1cm}$C = \{\emptyset, a\}$ 
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\item What is $B \times C$?
\item What is $\mid A \times A \mid$? 
\item Is $A \in B$?  Is $A \subseteq B$? 
\item What is $\mid A  \cup B  \cup C \mid$?
\item What is $P(C)$? 
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\item {[5 points]}   Let $A$, $B$, and $C$ be sets. Prove or disprove that $A - (B \cap C) = (A - B) \cup (A - C)$ using 
a membership table. State whether your membership table proves or disproves the statement and why. \\


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