\documentclass{article}   

\title{Discrete Mathematics\\ Drill-5} 
\author 
{Moshe Rosenfeld\\ 
\\ Hanoi 2011\\
{\tt moishe@u.washington.edu}}
\date{}
        

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\maketitle                

\section{Finite sets}
\begin{enumerate}
\item Construct a family of subsets of $\{1,2,3,4,5,6,7,8,9,10\}$ such that:
\begin{itemize}
\item Every subset has an even size.
\item Every two subsets have an even number of members in common (0 is even).
\item What is the largest number of such subsets you can construct?
\end{itemize}
\item 
Construct a family of subsets of $\{1,2,3,4,5,6,7,8,9,10\}$ such that:
\begin{itemize}
\item Every subset has an odd size.
\item Every two subsets have an even number of members in common (0 is even).
\item What is the largest number of such subsets you can construct?
\end{itemize}
\item Construct a family of subsets of $\{1,2,3,4,5\}$ such that every two subsets have exactly two nunbers in common.
\item How many elements does the set $P(P(P(\emptyset)))$ have?
\item Let $A = \{a,b,c,d,e\}$\\
\begin{enumerate}
\item What is the characteristic (incidence) vector of $\{a,b,e\}$?
\item What is the characteritic vector of $\{a,b,c,d\} \cap \{b,c,d,e\}$?
\item List all characteristic vectors of all subsets of $A$ with cardinality $4$.
\end{enumerate}
\item Let $S = \{1,2,\ldots, 7\}$ Find a family of $7$ triples such that any pair of distinct triples have exactly one number in common.
\item Prove that the $7$ characteristic vectors of the triples you found are linearly independent.
\item *Can you prove that it is not possible to find $8$ triples such that every two triples have exactly one number in common.
\end{enumerate}

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