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{\bf CSci 243 Homework 1}
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**My name**  %update me!!
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\begin{enumerate}

\item (4 points each) Construct a truth table for each of these compound propositions.
\begin{enumerate}
\item $(p\land q)\rightarrow (p\lor q)$
\item $((p\rightarrow q)\rightarrow r)\rightarrow s$
\item $(p \leftrightarrow  q)\oplus(p \leftrightarrow  \lnot q)$

\end{enumerate}

\item (4 points each) Using logical identities and laws, show the logic equivalence of
\begin{enumerate}
\item  $p\leftrightarrow q$ and $(p\land q)\lor(\lnot p\land \lnot q)$
\item  $p\leftrightarrow q$ and $\lnot p \leftrightarrow \lnot q$
\item  $(p\rightarrow r)\lor(q\rightarrow r)$ and $(p\land q)\rightarrow r$



\end{enumerate}


\item (4 points each) Recall the logic puzzle -- the island of knights and liars (knights only tell the truth, while liars always lie) -- we discussed in class. You encounter two people, A and B. Determine, if possible, what A and B are according to what they said. When there is no unique solution, list all possible solutions or state that there are no solutions. Explain your reasoning. 

\begin{enumerate}

\item A says "At least one of us is a liar" and B says nothing. 

\item A says "I am a liar or B is a knight" and B says nothing. 

\item Both A and B say "I am a knight."

\item Both A and B say "I am a liar."

\end{enumerate}


\item (\textbf{Bonus Question} 2 point\footnote{The points for bonus questions are added directly to the final grade.}) We want to determine the relative salaries of three coworkers -- A, B, C -- using two facts. First, we know that if A is not the highest paid of the three, then B is. Second we know that if B is not the lowest paid, then C is paid the most. Who is paid the most and who the least? Explain your reasoning. 


\end{enumerate}

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