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Solve the following problems.  Due Feb 5, 2009. 



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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).
  
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% 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.
  
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\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. 

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\item Prove that for all real numbers $a, b$ and $c$, 
\[   bc+ac+ab\le a^2+b^2+c^2.  \]

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\item Prove that for all real numbers $a$ and $b$, 
\[  |a|<|b| \Rightarrow a^2\le b^2.  \] 

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\item Prove that in  a graph the number of vertices with odd degrees is even. 
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