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\title{Math 214 Sample Homework}
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Solve the following problems. Due Feb 5, 2011.
\bigskip
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% taken from http://www.hku.hk/cgi-bin/philodep/knight/puzzle, puzzle 55
\item Recall the Knights and Knaves from the first day of class. Recall
that knights always tell the truth, and knaves always lie. You meet
three inhabitants: Patricia, Quinn and Roberta. Patricia claims that
it's false that Roberta is a knave. Quinn says, `Either Roberta is a
knight or I am a knight.' Roberta says that Quinn is a knave. Who are
knights and who are knaves? Prove your answer (using truth table).
\bigskip
% taken from http://www.hku.hk/cgi-bin/philodep/knight/puzzle, puzzle 110
\item Later, you meet Ann, Bert and Chuck. Ann says, `Chuck could claim
that I am a knight.' Bert says that only a knave would say that Ann is
a knave. Chuck claims, `Ann could say that I am a knave.' Who are
knights and who are knaves? Prove your answer.
\bigskip
\item By using truth tables prove that, for all statements P and Q, the statement `P$\Rightarrow$ Q' and `(not Q)$\Rightarrow$ (not P)' are equivalent.
\bigskip
\item Prove that for all real numbers $a, b$ and $c$,
\begin{equation*}
bc+ac+ab\le a^2+b^2+c^2.
\end{equation*}
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\item Prove that for all real numbers $a$ and $b$,
\begin{equation*}
|a|<|b| \Rightarrow a^2\le b^2.
\end{equation*}
\cite{shi}
\cite{shi1998singular}
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\item Write the set $A=\{-1,-2,-3,\cdots\}$ in a form $\{x\in {\mathbb Z}:p(x)\}$.
\end{enumerate}
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