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\begin{LARGE}
{\bf CSci 243 Homework 4}
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**My name**
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\begin{enumerate}

\item (7 points) Determine whether these statements are true or false.
\begin{enumerate}
\item $\varnothing \in \{ \varnothing \}$
\item $\varnothing\in\{\varnothing,\{\varnothing\}\}$
\item $\{\varnothing\}\in\{\varnothing\}$
\item $\{\varnothing\}\in\{\{\varnothing\}\}$
\item $\varnothing \subset \{\varnothing,\{\varnothing\}\}$
\item $\{\varnothing\}\subset\{\varnothing,\{\varnothing\}\}$
\item $\{\{\varnothing\}\}\subset\{\varnothing,\{\varnothing\}\}$
\end{enumerate}

\item (4 points) Is each of these sets the power set of a set, where $a$ and $b$ are distinct elements? If yes, give the original set.
\begin{enumerate}
\item $\varnothing$
\item $\{\varnothing,\{a\}\}$
\item $\{\varnothing,\{a\},\{\varnothing,a\}\}$
\item $\{\varnothing,\{a\},\{b\},\{a,b\}\}$
\end{enumerate} 


\item (10 points) For sets $A$, $B$, and $C$, prove that $(B-A)\cup(C-A)=(B\cup C)-A$
\begin{enumerate}
\item by showing each side is a subset of the other side
\item by using a membership table
\end{enumerate}

\item (5 points) Find these values.
\begin{enumerate}
\item $\lfloor 1.1\rfloor$
\item $\lceil 1.1\rceil$
\item $\lfloor -0.1\rfloor$
\item $\lceil -0.1\rceil$
\item $\lfloor \frac{1}{2}+\lceil \frac{1}{2}\rceil\rfloor$
\end{enumerate}

\item (8 points) Determine whether each of these functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ is one-to-one, onto, both, or neither.
\begin{enumerate}
\item $f(n)=n+1$
\item $f(n)=n^2+1$.
\item $f(n)=n^3$
\item $f(n)=\lceil\frac{n}{2}\rceil$
\end{enumerate}


\item (6 points) Prove that $\lceil x + n \rceil = \lceil x \rceil + n$, where $x$ is a real number and $n$ is an integer. (Hint: note the property of ceiling function: $\lceil x \rceil = n$ if and only if $n \leq x < n+1$).

\end{enumerate}

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