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\title{MA162: Finite mathematics}
\subtitle{Set Theory and Counting}
\author{Paul Koester}
\institute{University of Kentucky}
\date{February 12, 2014}

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\begin{document}

\maketitle

\begin{frame}[t]{Sets and Counting}
\begin{itemize}
  \item Jane has her MA 162 lecture on Monday and Wednesday and MA 162 recitation on Tuesday.
  \item She has her ACC 202 course on Monday, Wednesday, and Friday.
  \item She has her MGT 301 course on Monday evenings.
  \item In a given week, how many days does she have class?
\end{itemize}
\end{frame}

\begin{frame}[t]{}
\begin{itemize}
  \item Counting with numbers is only effective when things don't overlap. 
  \item We need to use \alert{sets} when there is the possibility of overlap.
  \item \(\{\mbox{Mon}, \mbox{Tues}, \mbox{Wed}\} \cup \{\mbox{Mon}, \mbox{Wed}, \mbox{Fri}\} \cup \{\mbox{Mon}\}
        = \{ \mbox{Mon}, \mbox{Tues}, \mbox{Wed}, \mbox{Fri}\} \)
\end{itemize}
\end{frame}

\begin{frame}[t]{Setting up some terminology}
\begin{itemize}
  \item A \alert{set} is a collection of objects. \vfill
  \item The objects in a set are called \alert{elements} of the set. \vfill
  \item ``x is an element of A'' is written as \(x \in A.\)
  \item These elements could be numbers, letters, names of people, places, functions, other sets, etc. \vfill
  \item The order of the elements does not matter.  \(\{1,2,3,4\}\) and \(\{4,2,1,3\}\) are identical as sets.\vfill
  \item Repetition does not matter.   \(\{1,2,3,4\}\) and \(\{1,2,3,3,3,4,1,2,2,1\}\) are identical as sets.\vfill
  \item The set with no elements is called the \alert{empty set} and is denoted \(\emptyset\)
\end{itemize}
\end{frame}

\begin{frame}[t]{Setting up some terminology}
\begin{itemize}
  \item A set A is a subset of the set B if every element of A is also an element of B. \vfill
  \item ``A is a subset of B'' is denoted \(A \subset B\). \vfill
  \item Let \(A = \{2,4,6\} \) and \(B = \{1,2,3,4,5,6\}\).  Then \(A \subset B\).
\end{itemize}
\end{frame}

\begin{frame}[t]{New sets from old}
\begin{itemize}
  \item The \alert{union} of A and B, denoted \(A \cup B\) is the set of all elements belonging to at least one of A or B.
  \item The \alert{intersection} of A and B, denoted \(A \cap B\) is the set of all elements belonging to both A and B.
  \item The \alert{difference} of A and B, denoted \(A - B\) is the set of all elements belonging to A but not B.
  \item Given a universal set U, the \alert{complement} of A, denoted \(A^c\) is the set of all elements of the 
    universal set which do not belong to A.
\end{itemize}
\end{frame}

\begin{frame}[t]{}
\begin{itemize}
  \item Coin is flipped three times; 
  \item \(A = \{HHH, HHT, HTH, HTT\} \) = Heads on first flip
  \item \(B = \{HHT, HTH, THH\}\) = exactly 2 heads.\vfill
  \item \(A \cup B \) = \vfill
  \item \(A \cap B \) = \vfill
  \item \(A - B\) = \vfill
  \item \(B - A\) = \vfill
  \item \(A^c\) = \vfill
  \item \(B^c\) = \vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{De Morgan's Laws}
If A and B are two sets, then 
\[ (A \cup B)^c = A^c \cap B^c\]
and 
\[ (A \cap B)^c = A^c \cup B^c\]
\end{frame}

\begin{frame}[t]{}
For the coin flipping problem find each of 
\begin{itemize}
  \item \( ( A \cup B)^c \) \vfill
  \item \( A^c \cap B^c \)  \vfill
  \item \( A^c \cup B^c \)  \vfill
  \item \( (A \cap B)^c \)  \vfill
\end{itemize}
\end{frame}

\begin{frame}[t]{To include or not to include...}
Consider a standard deck of 52 playing cards.  \\
How many cards are either red or have face value equal to 10? \\

\end{frame}

\begin{frame}[t]{Inclusion-Exclusion Principle}
\begin{itemize}
  \item \textbf{Two set inclusion-exclusion:} 
    \[n(A \cup B) = n(A) + n(B) - n(A\cap B)\]
  \item \textbf{Three set inclusion-exclusion:}  
    \[n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A\cap B) - n(A \cap C)\] \[ - n(B \cap C) + n(A\cap B \cap C)\]
  \item \textbf{Many set inclusion-exclusion:}  The above two versions suggest a pattern for four sets, five sets, etc.
    The idea isn't too hard, but developing enough notation to write down a general formula is a bit of a challenge...
\end{itemize}
\end{frame}

\begin{frame}[t]{}
200 people were surveyed about dining preferences.  155 people surveyed like Mexican food.  
75 people surveyed like Thai food.  20 of the people surveyed like neither Mexican food nor Thai food.
How many of the surveyed people like Mexican food but not Thai food?     
\end{frame}

\begin{frame}[t]{To include or not to include...}
100 students were interviewed.
\begin{itemize} 
  \item 58 of them have part-time jobs.
  \item 43 of them are involved in extra-curricular activities at school.
  \item 25 of them are involved in community outreach programs.
  \item 28 of them have part-time jobs and are involved in extra-curricular activities
  \item 10 of them have part-time jobs are are involved in community outreach programs
  \item 13 of them are involved in extra-curricular activities and community outreach programs.
  \item Each student interviewed was involved in at least one of the above three activities.
  \item How many students from this group were involved in all three of the above activities?
\end{itemize}
\end{frame}

\begin{frame}[t]{To include or not to include...}
Roll a pair of fair six sided dice with the digits 1,2,3,4,5,6.  \\
How many outcomes have either at least one die showing a 4 or the sum of the dice is equal to 9  \\

\end{frame}



\end{document}

