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%========================================================================
\begin{center}
\textbf{\Large{Homework 10}} \vspace{2mm}

Mathematics in Computer Science
%John E. Hopcroft
\end{center}
\vspace{1mm}

\begin{enumerate}
  \item What are the Boolean functions
    \begin{enumerate}
      \item $(x\rightarrow y)\rightarrow y$?
      \item $(x\rightarrow y)\rightarrow x$?
    \end{enumerate}
 %
 \item There are 16 Boolean functions of two variables.  A subset of
 the 16 Boolean functions is said to be \emph{universal} if the other
  functions can be expressed in terms of functions in the subset.  A
 subset is \emph{minimal} if deleting any element of the subset results in a
non universal set. `AND' and `NOT' is a minimal universal set as is `OR'
and `NOT'.  

Give three other such sets.  Demonstrate that your sets
are indeed universal.
%
   \item  How many Boolean functions of $n$ variables?
%
   \item There are two types of people in a certain group.  Liars who
   always lie and truth tellers who always say the truth.  In each of
   the following can you say anything about the type of each person?
  \begin{enumerate}
      \item $A$ says one of us is a liar.  $B$ says nothing.
       
     \item $A$ says both of us are truth tellers and $B$ says $A$ is a
     liar.
    
     \item $A$ and $B$ both say we are truth tellers.
 
  \end{enumerate}
%
  \item Prove that $x\rightarrow
y$ is equivalent to $\overline{y}\rightarrow\overline{x}$

\item  Aces and eights - 8 cards 4 aces 4 eights, three players, six
 cards dealt, two to each player. Other two face down.  Each layer
 show his or her hand to the other two players.  In turn each player
 tries to determine his or her hand.  If player does not know his or
 her hand he/she must say so.
 \begin{enumerate}
 \item Alice two aces, Bob two eights, you.

~Alice don't know, Bob don't know.  Then what are you holding?  
 \item you, Alice two eights, Bob ace~+~eight.
 
~You don't know, Alice don't know, Bob don't know. Then what are you holding?
 \end{enumerate}
 %
\end{enumerate}


%========================================================================
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