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\title{CS200 - Problem Set 2\\\small{Due: Monday, Sep. 25 to Canvas before class}}
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Please read the sections of the syllabus on problem sets and honor code before starting this homework.
\begin{enumerate}
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\item \textit{Set Builder to Roster Notation}
\textbf{[2 point each]}
The following sets are described in set builder notation.
Describe each of them in roster notation, instead.
The following symbols are used:
$\mathbb{Z}$ denotes the set of integers; $\mathbb{R}$ denotes the set of real numbers;
$\mathbb{N}$ denotes the set of natural numbers, i.e., $\mathbb{N} = \{0, 1, 2, \dots \}$.
\begin{enumerate}
\item
$\{r \mid r \in \mathbb{R} \mbox{ and } r = r^2 \}$
\item
$\{n \mid n \in \mathbb{N} \mbox{ and } n > n^2 \}$
\item
$\{x \mid \mbox{$x$ is a letter in the word \emph{accommodate}}\}$
\item
$\{z^2 \mid z \in \mathbb{Z} \mbox{ and } 6 < z^3 < 160 \}$.
\item
$\{S \subseteq \{2, 4, 6, 8\} \mid S \cap \{2,4\} \neq \emptyset \mbox{ and $|S|$ is even} \}$
\end{enumerate}
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\item \textit{Set builder notation} \textbf{[3 points each]}\\ Write each of the following sets using set-builder notation:
\begin{enumerate}
\item $\{\dots,1/8,1/6,1/4,1/2,2,4,6,8\dots\}$
\item Express the set of all sets of 2 integers such that the two numbers in the set are non-zero, have opposite signs,
and the magnitude of one of them is the square of the magnitude of the other.
\end{enumerate}
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\item \textit{Universal Set} \textbf{[2 points]}\\
Let $U$, the universal set, be the set of even integers from 2 to 12 inclusive, and
let $A = \{4, 6, 7, 9\}$, $B = \{2, 3, 4, 5, 7\}$. What is $\overline{A - B}$?
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\item \textit{Set Operations} \textbf{[2 points each]}
%(3 points)
Simplify each of the following expressions, where $A$ is an arbitrary set, $\emptyset$ is the empty set, and $U$ is the universal set. Hint: each answer to (a)-(h) is one of $A$, $U$, or $\emptyset$.
Just write down the answer: no proof needed, no steps need be shown.
\begin{enumerate}
\item
$A \cap U$
\item
$A \cap \emptyset$
\item
$A \cup U$
\item
$A \cup \emptyset$
\item
$A \cup A$
\item
$A \cap A$
\item
$A \cup \overline{A}$
\item
$A \cap \overline{A}$
\end{enumerate}
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\item \textit{Statements} \textbf{[3 points each]} For each of the following sentences, decide whether it is a statement, predicate, or neither, and explain why
\begin{enumerate}
\item Call me Ishmael.
\item The universe is supported on the back of a giant tortoise.
\item $x$ is a multiple of $7.$
\item The next sentence is true.
\item The preceding sentence is false.
\item The set $\mathbb{Z}$ contains an infinite number of elements.
\end{enumerate}
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\item\textit{Statements} \textbf{[2 point each]}
% (4 points)
Simplify each of the following expressions, where $p$ denotes a statement, and $T$ and $F$ are the
Boolean constants {\em true} and {\em false}. Hint: each answer is one of $p$, $T$, or $F$. No proof needed, no steps need be shown.
\begin{enumerate}
\item
$T \wedge p$
\item
$F \wedge p$
\item
$T \vee p$
\item
$F \vee p$
\item
$p \vee p$
\item
$p \wedge p$
\item
$p \vee \neg p$
\end{enumerate}
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\item How long did you spend on this homework?
\end{enumerate}
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