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\title{On Examples, Counterexamples, and Proof by Example}
\author{Thomas Mitchell}
\begin{document}
\maketitle

This piece is motivated by two seemingly unrelated items that have recently appeared in the \textit{Journal} (\cite{bib:Dudley} and \cite{bib:Morrill}). In \cite{bib:Dudley}, the editor of the \textit{Journal} wrote of ``mathematical cranks'' and how frustratingly difficult it can be sometimes to get would-be ``circle-squarers'' and ``angle-trisectors'' to see the errors of their ways.  As an economics professor with a degree in mathematics, I am only too well aware of circle-squarers, angle-trisectors, and those who would make implicit use of the ``Law of Universal Cancellation,'' also mentioned in \cite{bib:Dudley}.

\section*{On the ``Law of Universal Cancellation''}
A few years ago I wrote a short paper (\cite{bib:Mitchell}) that I hoped would discourage economics students from employing a proof ``technique'' that I call ``proof by example.''  Sometimes objectives are best obtained through humor, so consider the following ``theorem,'' which formalizes the Law of Universal Cancellation.
\begin{mythm}
\label{thm:cancel}
If the numerator and denominator of a fraction have a digit in common, then the fraction can be reduced by simply striking the common digit from the numerator and denominator.
\end{mythm}

\noindent \textbf{``Proof.''}
Clearly, the theorem holds if the numerator and denominator both end in ``zero''; we can strike the trailing zero from both the numerator and the denominator and correctly reduce the fraction.  Less obviously, consider the following:
\[ \begin{array}{cc cc}
     \frac{\textstyle 1/\!\!\!6}{\textstyle /\!\!\!64} =
          \frac{\textstyle 1}{\textstyle 4} , &
     \frac{\textstyle 1/\!\!\!9}{\textstyle /\!\!\!95} =
          \frac{\textstyle 1}{\textstyle 5} , &
     \frac{\textstyle 2/\!\!\!6}{\textstyle /\!\!\!65} =
          \frac{\textstyle 2}{\textstyle 5} , &
     \frac{\textstyle 4/\!\!\!9}{\textstyle /\!\!\!98} =
          \frac{\textstyle 4}{\textstyle 8} ; \\
     \hdotsfor{4} \\
     \frac{\textstyle /\!\!\!34}{\textstyle 1/\!\!\!36} =
          \frac{\textstyle 4}{\textstyle 16} , &
     \frac{\textstyle /\!\!\!34}{\textstyle 2/\!\!\!38} =
          \frac{\textstyle 4}{\textstyle 28} , &
     \frac{\textstyle /\!\!\!67}{\textstyle 2/\!\!\!68} =
          \frac{\textstyle 7}{\textstyle 28} , &
     \frac{\textstyle /\!\!\!67}{\textstyle 4/\!\!\!69} =
          \frac{\textstyle 7}{\textstyle 49} , \\[5pt]
     \frac{\textstyle /\!\!\!96}{\textstyle 1/\!\!\!92} =
          \frac{\textstyle 6}{\textstyle 12} , &
     \frac{\textstyle /\!\!\!97}{\textstyle 1/\!\!\!94} =
          \frac{\textstyle 7}{\textstyle 14} , &
     \frac{\textstyle /\!\!\!97}{\textstyle 2/\!\!\!91} =
          \frac{\textstyle 7}{\textstyle 21} , &
     \frac{\textstyle /\!\!\!98}{\textstyle 1/\!\!\!96} =
          \frac{\textstyle 8}{\textstyle 16} , \\[5pt]
     &
     \frac{\textstyle /\!\!\!98}{\textstyle 2/\!\!\!94} =
          \frac{\textstyle 8}{\textstyle 24} , &
     \frac{\textstyle /\!\!\!98}{\textstyle 3/\!\!\!92} =
          \frac{\textstyle 8}{\textstyle 32} ; & \\
     \hdotsfor{4} \end{array} \]
\[ \begin{array}{llll}
     \frac{\textstyle /\!\!\!23}{\textstyle 1/\!\!\!265} =
          \frac{\textstyle 3}{\textstyle 165} , &
     \frac{\textstyle /\!\!\!46}{\textstyle 1/\!\!\!495} =
          \frac{\textstyle 6}{\textstyle 195} , &
     \frac{\textstyle /\!\!\!54}{\textstyle 1/\!\!\!512} =
          \frac{\textstyle 4}{\textstyle 112} , &
     \frac{\textstyle /\!\!\!56}{\textstyle 4/\!\!\!592} =
          \frac{\textstyle 6}{\textstyle 492} , \\[5pt]
     \frac{\textstyle /\!\!\!57}{\textstyle 1/\!\!\!596} =
          \frac{\textstyle 7}{\textstyle 196} , &
     \frac{\textstyle /\!\!\!67}{\textstyle 3/\!\!\!685} =
          \frac{\textstyle 7}{\textstyle 385} , &
     \frac{\textstyle 2/\!\!\!67}{\textstyle 14/\!\!\!685} =
          \frac{\textstyle 27}{\textstyle 1485} , &
     \frac{\textstyle 5/\!\!\!01}{\textstyle 39/\!\!\!078} =
          \frac{\textstyle 51}{\textstyle 3978} , \\[5pt]
     \frac{\textstyle 6/\!\!\!12}{\textstyle 50/\!\!\!184} =
          \frac{\textstyle 62}{\textstyle 5084} , &
     \frac{\textstyle 8/\!\!\!46}{\textstyle 27/\!\!\!495} =
          \frac{\textstyle 86}{\textstyle 2795} , &
     \frac{\textstyle 9/\!\!\!34}{\textstyle 72/\!\!\!385} =
          \frac{\textstyle 94}{\textstyle 7285} . \end{array} \]
In the following, simply delete the zero in the middle:
\[ \begin{array}{lllll}
     \frac{\textstyle 101}{\textstyle 202} =
     \frac{\textstyle 11}{\textstyle 22} , &
          \frac{\textstyle 101}{\textstyle 303} =
               \frac{\textstyle 11}{\textstyle 33} , &
          \cdots , &
          \frac{\textstyle 101}{\textstyle 808} =
               \frac{\textstyle 11}{\textstyle 88} , &
          \frac{\textstyle 101}{\textstyle 909} =
               \frac{\textstyle 11}{\textstyle 99} , \\[5pt]
\mbox{ } & \frac{\textstyle 202}{\textstyle 303} =
     \frac{\textstyle 22}{\textstyle 33} , &
          \cdots , &
          \frac{\textstyle 202}{\textstyle 808} =
               \frac{\textstyle 22}{\textstyle 88} , &
          \frac{\textstyle 202}{\textstyle 909} =
               \frac{\textstyle 22}{\textstyle 99} , \\[5pt]
\mbox{ } & \mbox{ } & \ddots   & \vdots  &  \vdots \\[5pt]
\mbox{ } & \mbox{ } & \mbox{ } &
          \frac{\textstyle 707}{\textstyle 808} =
               \frac{\textstyle 77}{\textstyle 88} , &
          \frac{\textstyle 707}{\textstyle 909} =
               \frac{\textstyle 77}{\textstyle 99} , \\[5pt]
\mbox{ } & \mbox{ } & \mbox{ } & \mbox{ } &
          \frac{\textstyle 808}{\textstyle 909} =
               \frac{\textstyle 88}{\textstyle 99} ; \\
              \hdotsfor{5} \end{array} \]
In the following, the middle digits can be deleted to reduce the fraction:
\[ \begin{array}{lll lll}
    \frac{\textstyle 134}{\textstyle 737} =
          \frac{\textstyle 14}{\textstyle 77} , &
    \frac{\textstyle 143}{\textstyle 242} =
          \frac{\textstyle 13}{\textstyle 22} , &
    \frac{\textstyle 156}{\textstyle 858} =
          \frac{\textstyle 16}{\textstyle 88} , &
    \frac{\textstyle 165}{\textstyle 363} =
          \frac{\textstyle 15}{\textstyle 33} , &
    \frac{\textstyle 178}{\textstyle 979} =
          \frac{\textstyle 18}{\textstyle 99} , &
    \frac{\textstyle 187}{\textstyle 484} =
          \frac{\textstyle 17}{\textstyle 44} , \\[5pt]
    \frac{\textstyle 242}{\textstyle 341} =
          \frac{\textstyle 22}{\textstyle 31} , &
    \frac{\textstyle 264}{\textstyle 363} =
          \frac{\textstyle 24}{\textstyle 33} , &
    \frac{\textstyle 286}{\textstyle 484} =
          \frac{\textstyle 26}{\textstyle 44} , &
    \frac{\textstyle 363}{\textstyle 462} =
          \frac{\textstyle 33}{\textstyle 42} , &
    \frac{\textstyle 363}{\textstyle 561} =
          \frac{\textstyle 33}{\textstyle 51} , &
    \frac{\textstyle 385}{\textstyle 484} =
          \frac{\textstyle 35}{\textstyle 44} , \\[5pt]
    \frac{\textstyle 484}{\textstyle 583} =
          \frac{\textstyle 44}{\textstyle 53} , &
    \frac{\textstyle 484}{\textstyle 682} =
          \frac{\textstyle 44}{\textstyle 62} , &
    \frac{\textstyle 484}{\textstyle 781} =
          \frac{\textstyle 44}{\textstyle 71} , &
    \frac{\textstyle 536}{\textstyle 737} =
          \frac{\textstyle 56}{\textstyle 77} , &
    \frac{\textstyle 737}{\textstyle 938} =
          \frac{\textstyle 77}{\textstyle 98} ; \\[8.25pt]
    \frac{\textstyle 102}{\textstyle 306} =
          \frac{\textstyle 12}{\textstyle 36} , &
    \frac{\textstyle 102}{\textstyle 408} =
          \frac{\textstyle 12}{\textstyle 48} , &
    \frac{\textstyle 103}{\textstyle 206} =
          \frac{\textstyle 13}{\textstyle 26} , &
    \frac{\textstyle 104}{\textstyle 208} =
          \frac{\textstyle 14}{\textstyle 28} , &
    \frac{\textstyle 134}{\textstyle 536} =
          \frac{\textstyle 14}{\textstyle 56} , &
    \frac{\textstyle 134}{\textstyle 938} =
          \frac{\textstyle 14}{\textstyle 98} , \\[5pt]
    \frac{\textstyle 136}{\textstyle 238} =
          \frac{\textstyle 16}{\textstyle 28} , &
    \frac{\textstyle 154}{\textstyle 253} =
          \frac{\textstyle 14}{\textstyle 23} , &
    \frac{\textstyle 154}{\textstyle 352} =
          \frac{\textstyle 14}{\textstyle 32} , &
    \frac{\textstyle 165}{\textstyle 264} =
          \frac{\textstyle 15}{\textstyle 24} , &
    \frac{\textstyle 165}{\textstyle 462} =
          \frac{\textstyle 15}{\textstyle 42} , &
    \frac{\textstyle 176}{\textstyle 275} =
          \frac{\textstyle 16}{\textstyle 25} , \\[5pt]
    \frac{\textstyle 176}{\textstyle 374} =
          \frac{\textstyle 16}{\textstyle 34} , &
    \frac{\textstyle 176}{\textstyle 473} =
          \frac{\textstyle 16}{\textstyle 43} , &
    \frac{\textstyle 176}{\textstyle 572} =
          \frac{\textstyle 16}{\textstyle 52} , &
    \frac{\textstyle 187}{\textstyle 286} =
          \frac{\textstyle 17}{\textstyle 26} , &
    \frac{\textstyle 187}{\textstyle 385} =
          \frac{\textstyle 17}{\textstyle 35} , &
    \frac{\textstyle 187}{\textstyle 583} =
          \frac{\textstyle 17}{\textstyle 53} , \\[5pt]
    \frac{\textstyle 187}{\textstyle 682} =
          \frac{\textstyle 17}{\textstyle 62} , &
    \frac{\textstyle 195}{\textstyle 390} =
          \frac{\textstyle 15}{\textstyle 30} , &
    \frac{\textstyle 196}{\textstyle 294} =
          \frac{\textstyle 16}{\textstyle 24} , &
    \frac{\textstyle 196}{\textstyle 392} =
          \frac{\textstyle 16}{\textstyle 32} , &
    \frac{\textstyle 196}{\textstyle 490} =
          \frac{\textstyle 16}{\textstyle 40} , &
    \frac{\textstyle 197}{\textstyle 394} =
          \frac{\textstyle 17}{\textstyle 34} , \\[5pt]
    \frac{\textstyle 198}{\textstyle 297} =
          \frac{\textstyle 18}{\textstyle 27} , &
    \frac{\textstyle 198}{\textstyle 396} =
          \frac{\textstyle 18}{\textstyle 36} , &
    \frac{\textstyle 198}{\textstyle 495} =
          \frac{\textstyle 18}{\textstyle 45} , &
    \frac{\textstyle 198}{\textstyle 594} =
          \frac{\textstyle 18}{\textstyle 54} , &
    \frac{\textstyle 198}{\textstyle 693} =
          \frac{\textstyle 18}{\textstyle 63} , &
    \frac{\textstyle 198}{\textstyle 792} =
          \frac{\textstyle 18}{\textstyle 72} , \\[5pt]
    \frac{\textstyle 201}{\textstyle 603} =
          \frac{\textstyle 21}{\textstyle 63} , &
    \frac{\textstyle 201}{\textstyle 804} =
          \frac{\textstyle 21}{\textstyle 84} , &
    \frac{\textstyle 203}{\textstyle 406} =
          \frac{\textstyle 23}{\textstyle 46} , &
    \frac{\textstyle 203}{\textstyle 609} =
          \frac{\textstyle 23}{\textstyle 69} , &
    \frac{\textstyle 204}{\textstyle 306} =
          \frac{\textstyle 24}{\textstyle 36} , &
    \frac{\textstyle 206}{\textstyle 309} =
          \frac{\textstyle 26}{\textstyle 39} , \\[5pt]
    \frac{\textstyle 234}{\textstyle 936} =
          \frac{\textstyle 24}{\textstyle 96} , &
    \frac{\textstyle 253}{\textstyle 451} =
          \frac{\textstyle 23}{\textstyle 41} , &
    \frac{\textstyle 264}{\textstyle 561} =
          \frac{\textstyle 24}{\textstyle 51} , &
    \frac{\textstyle 268}{\textstyle 469} =
          \frac{\textstyle 28}{\textstyle 49} , &
    \frac{\textstyle 275}{\textstyle 374} =
          \frac{\textstyle 25}{\textstyle 34} , &
    \frac{\textstyle 275}{\textstyle 473} =
          \frac{\textstyle 25}{\textstyle 43} , \\[5pt]
    \frac{\textstyle 275}{\textstyle 671} =
          \frac{\textstyle 25}{\textstyle 61} , &
    \frac{\textstyle 286}{\textstyle 385} =
          \frac{\textstyle 26}{\textstyle 35} , &
    \frac{\textstyle 286}{\textstyle 583} =
          \frac{\textstyle 26}{\textstyle 53} , &
    \frac{\textstyle 286}{\textstyle 781} =
          \frac{\textstyle 26}{\textstyle 71} , &
    \frac{\textstyle 297}{\textstyle 396} =
          \frac{\textstyle 27}{\textstyle 36} , &
    \frac{\textstyle 297}{\textstyle 495} =
          \frac{\textstyle 27}{\textstyle 45} , \\[5pt]
    \frac{\textstyle 297}{\textstyle 594} =
          \frac{\textstyle 27}{\textstyle 54} , &
    \frac{\textstyle 297}{\textstyle 693} =
          \frac{\textstyle 27}{\textstyle 63} , &
    \frac{\textstyle 297}{\textstyle 891} =
          \frac{\textstyle 27}{\textstyle 81} , &
    \frac{\textstyle 298}{\textstyle 596} =
          \frac{\textstyle 28}{\textstyle 56} , &
    \frac{\textstyle 301}{\textstyle 602} =
          \frac{\textstyle 31}{\textstyle 62} , &
    \frac{\textstyle 302}{\textstyle 604} =
          \frac{\textstyle 32}{\textstyle 64} , \\[5pt]
    \frac{\textstyle 302}{\textstyle 906} =
          \frac{\textstyle 32}{\textstyle 96} , &
    \frac{\textstyle 304}{\textstyle 608} =
          \frac{\textstyle 34}{\textstyle 68} , &
    \frac{\textstyle 306}{\textstyle 408} =
          \frac{\textstyle 36}{\textstyle 48} , &
    \frac{\textstyle 352}{\textstyle 451} =
          \frac{\textstyle 32}{\textstyle 41} , &
    \frac{\textstyle 374}{\textstyle 572} =
          \frac{\textstyle 34}{\textstyle 52} , &
    \frac{\textstyle 374}{\textstyle 671} =
          \frac{\textstyle 34}{\textstyle 61} , \\[5pt]
    \frac{\textstyle 385}{\textstyle 682} =
          \frac{\textstyle 35}{\textstyle 62} , &
    \frac{\textstyle 385}{\textstyle 781} =
          \frac{\textstyle 35}{\textstyle 71} , &
    \frac{\textstyle 392}{\textstyle 490} =
          \frac{\textstyle 32}{\textstyle 40} , &
    \frac{\textstyle 394}{\textstyle 591} =
          \frac{\textstyle 34}{\textstyle 51} , &
    \frac{\textstyle 395}{\textstyle 790} =
          \frac{\textstyle 35}{\textstyle 70} , &
    \frac{\textstyle 396}{\textstyle 495} =
          \frac{\textstyle 36}{\textstyle 45} , \\[5pt]
    \frac{\textstyle 396}{\textstyle 594} =
          \frac{\textstyle 36}{\textstyle 54} , &
    \frac{\textstyle 396}{\textstyle 792} =
          \frac{\textstyle 36}{\textstyle 72} , &
    \frac{\textstyle 396}{\textstyle 891} =
          \frac{\textstyle 36}{\textstyle 81} , &
    \frac{\textstyle 398}{\textstyle 597} =
          \frac{\textstyle 38}{\textstyle 57} , &
    \frac{\textstyle 398}{\textstyle 796} =
          \frac{\textstyle 38}{\textstyle 76} , &
    \frac{\textstyle 401}{\textstyle 802} =
          \frac{\textstyle 41}{\textstyle 82} , \\[5pt]
    \frac{\textstyle 402}{\textstyle 603} =
          \frac{\textstyle 42}{\textstyle 63} , &
    \frac{\textstyle 403}{\textstyle 806} =
          \frac{\textstyle 43}{\textstyle 86} , &
    \frac{\textstyle 462}{\textstyle 561} =
          \frac{\textstyle 42}{\textstyle 51} , &
    \frac{\textstyle 473}{\textstyle 572} =
          \frac{\textstyle 43}{\textstyle 52} , &
    \frac{\textstyle 473}{\textstyle 671} =
          \frac{\textstyle 43}{\textstyle 61} , &
    \frac{\textstyle 495}{\textstyle 693} =
          \frac{\textstyle 45}{\textstyle 63} , \\[5pt]
    \frac{\textstyle 495}{\textstyle 792} =
          \frac{\textstyle 45}{\textstyle 72} , &
    \frac{\textstyle 495}{\textstyle 891} =
          \frac{\textstyle 45}{\textstyle 81} , &
    \frac{\textstyle 532}{\textstyle 931} =
          \frac{\textstyle 52}{\textstyle 91} , &
    \frac{\textstyle 536}{\textstyle 938} =
          \frac{\textstyle 56}{\textstyle 98} , &
    \frac{\textstyle 572}{\textstyle 671} =
          \frac{\textstyle 52}{\textstyle 61} , &
    \frac{\textstyle 583}{\textstyle 682} =
          \frac{\textstyle 53}{\textstyle 62} , \\[5pt]
    \frac{\textstyle 583}{\textstyle 781} =
          \frac{\textstyle 53}{\textstyle 71} , &
    \frac{\textstyle 594}{\textstyle 693} =
          \frac{\textstyle 54}{\textstyle 63} , &
    \frac{\textstyle 594}{\textstyle 792} =
          \frac{\textstyle 54}{\textstyle 72} , &
    \frac{\textstyle 594}{\textstyle 891} =
          \frac{\textstyle 54}{\textstyle 81} , &
    \frac{\textstyle 596}{\textstyle 894} =
          \frac{\textstyle 56}{\textstyle 84} , &
    \frac{\textstyle 602}{\textstyle 903} =
          \frac{\textstyle 62}{\textstyle 93} , \\[5pt]
    \frac{\textstyle 603}{\textstyle 804} =
          \frac{\textstyle 63}{\textstyle 84} , &
    \frac{\textstyle 682}{\textstyle 781} =
          \frac{\textstyle 62}{\textstyle 71} , &
    \frac{\textstyle 693}{\textstyle 792} =
          \frac{\textstyle 63}{\textstyle 72} , &
    \frac{\textstyle 693}{\textstyle 891} =
          \frac{\textstyle 63}{\textstyle 81} , &
    \frac{\textstyle 792}{\textstyle 891} =
          \frac{\textstyle 72}{\textstyle 81} ; \end{array} \]

\[ \begin{array}{cc c cc}
    \frac{\textstyle 2/\!\!\!65}{\textstyle 10/\!\!\!6} =
          \frac{\textstyle 25}{\textstyle 10} , &
    \frac{\textstyle 2/\!\!\!98}{\textstyle 14/\!\!\!9} =
          \frac{\textstyle 28}{\textstyle 14} , &
    \frac{\textstyle 3/\!\!\!65}{\textstyle 14/\!\!\!6} =
          \frac{\textstyle 35}{\textstyle 14} , &
    \frac{\textstyle 4/\!\!\!65}{\textstyle 18/\!\!\!6} =
          \frac{\textstyle 45}{\textstyle 18} , &
    \frac{\textstyle 5/\!\!\!96}{\textstyle 14/\!\!\!9} =
          \frac{\textstyle 56}{\textstyle 14} , \\[5pt]
    \frac{\textstyle 6/\!\!\!95}{\textstyle 13/\!\!\!9} =
          \frac{\textstyle 65}{\textstyle 13} , &
    \frac{\textstyle 6/\!\!\!98}{\textstyle 34/\!\!\!9} =
          \frac{\textstyle 68}{\textstyle 34} , &
    \frac{\textstyle 7/\!\!\!32}{\textstyle 18/\!\!\!3} =
          \frac{\textstyle 72}{\textstyle 18} , &
    \frac{\textstyle 7/\!\!\!65}{\textstyle 30/\!\!\!6} =
          \frac{\textstyle 75}{\textstyle 30} , &
    \frac{\textstyle 8/\!\!\!54}{\textstyle 30/\!\!\!5} =
          \frac{\textstyle 84}{\textstyle 30} , \\[5pt]
    \frac{\textstyle 8/\!\!\!64}{\textstyle 21/\!\!\!6} =
          \frac{\textstyle 84}{\textstyle 21} , &
    \frac{\textstyle 8/\!\!\!65}{\textstyle 34/\!\!\!6} =
          \frac{\textstyle 85}{\textstyle 34} , &
    \frac{\textstyle 8/\!\!\!95}{\textstyle 17/\!\!\!9} =
          \frac{\textstyle 85}{\textstyle 17} , &
    \frac{\textstyle 9/\!\!\!65}{\textstyle 38/\!\!\!6} =
          \frac{\textstyle 95}{\textstyle 38} , &
    \frac{\textstyle 9/\!\!\!76}{\textstyle 42/\!\!\!7} =
          \frac{\textstyle 96}{\textstyle 42} \ldots \end{array} \]
The list could go on and on, so surely the theorem is correct. $\quad$ \textbf{Q.E.D.}

Of course the theorem is false, as a single counterexample demonstrates: $13/35 \neq 1/5$.  Is it not a beautiful demonstration of the elegance of mathematical reasoning that all of the examples above are insufficient to establish the truth of the ``theorem,'' yet a single, simple counterexample conclusively establishes its falsity?

\section*{A True Theorem}
In elementary set theory, the binary operation of ``set difference'' removes those elements from the first set that are also elements in the second set:  $A \setminus B = \{ x: x \in A \text{ and } x \not\in B \}$.  Readers of the \textit{Journal} should be familiar with the binary operation of set ``intersection'':  the intersection of two sets is the set of all things that are elements in \textit{both} sets:  
$A \bigcap B = \{ x: x \in A \text{ and } x \in B \}$.  Utilizing these two binary operations, the following theorem is true.
\begin{mythm}
\label{thm:disjoint}
If $A$ and $B$ are nonempty sets, then
$A \setminus B$, $B \setminus A$, and $A \bigcap B$ are pairwise disjoint.
\end{mythm}
That is, $(A \setminus B) \bigcap (B \setminus A) =
          (A \setminus B) \bigcap (A \bigcap B)   =
          (B \setminus A) \bigcap (A \bigcap B) = \varnothing$.

I have given the proof of theorem~\ref{thm:disjoint} as homework many times and many times I have read ``proofs'' that begin, 
``Let $A = \{ 1, 2, 3, 4 \}$ and $B = \{ 3, 4, 5, 6 \}$ {\ldots }''
This is what I call ``proof by example.'' ``Proof by example'' is just as invalid for establishing a true theorem like theorem~\ref{thm:disjoint} as it is invalid for establishing a false theorem like theorem~\ref{thm:cancel}.  The problem with ``proof by example,'' of course, is that one may unintentionally, or unwittingly, choose examples that are ``special cases'' in one or more ways.  In geometry, for example, it would not be appropriate to begin a proof of a theorem about triangles with, ``Let $\bigtriangleup ABC$ be an isosceles triangle with $AB = BC${\ldots}''\@  Even worse---but probably even easier to prove!---would be a ``proof'' beginning with ``Let $\bigtriangleup ABC$ be an equilateral triangle{\ldots}''\@    We should not study the equilateral triangle to establish general propositions about all triangles because equilateral triangles possess properties and characteristics not shared by all triangles.  It turns out that the equilateral triangle is to the study of geometry what the ``Cobb-Douglas utility function'' is to the study of consumer behavior in economics.  The Cobb-Douglas functional form possesses properties and characteristics not shared by all types of functions that may be used as a utility function.  This brings us to the results derived in \cite{bib:Morrill}.

\section*{On the Cobb-Douglas Utility Function}
Suppose that a consumer has already made the labor-leisure decision discussed in \cite{bib:Morrill}, and that as a consequence of that decision the consumer-worker chooses to work $h = 50$~hours per week (use 60~hours for a professor and 70~hours for an untenured professor\ldots).  If the consumer-worker earns an hourly wage of $w$, then the consumer can expect a weekly income of $50w$~dollars, out of which s/he chooses those amounts of food ($F$), clothing ($C$), and shelter ($S$) that will make her/him best off.  (Let shelter include any applicable utilities; e.g., gas, water, phone, electricity, cable TV.)  If the consumer faces food, clothing, and shelter prices of $p_F$, $p_C$, and $p_S$, respectively, and if we let $I$ denote the consumer's weekly income ($I = 50w$), then the consumer's problem is to maximize utility subject to a budget constraint
\[ p_F \cdot F + p_C \cdot C + p_S \cdot S = I . \]
Making use of the Cobb-Douglas utility function, let the consumer's utility function be $U(F, C, S) = F^{a_F} C^{a_C} S^{a_S}$.  The consumer's decision can now be formulated mathematically,
\[ \max_{F, C, S} \; F^{a_F} C^{a_C} S^{a_S} \; \text{ subject to } \; 
   p_F \cdot F + p_C \cdot C + p_S \cdot S = I . \]

Elementary calculus generates the following expressions for the optimal quantities of food, clothing, and shelter, given in terms of the problem's parameters, namely the prices of food, clothing, and shelter, the consumer's weekly income, and the exponents in the Cobb-Douglas utility function: 
\begin{equation}
\label{eq:my_F/c/s}
   F^* = \frac{a_F}{a_F + a_C + a_S} \cdot \frac{I}{p_F}, \quad
   C^* = \frac{a_C}{a_F + a_C + a_S} \cdot \frac{I}{p_C}, \quad
   S^* = \frac{a_S}{a_F + a_C + a_S} \cdot \frac{I}{p_S}.
\end{equation}
These functions are what economists call ``ordinary demand functions''; each of them would be plotted as a rectangular hyperbola in their own  coordinate system just as in figure~1 of \cite{bib:Morrill}.  Expressed in a different way,
\[ p_F \cdot F^* = \frac{a_F}{a_F + a_C + a_S} I, \quad
   p_C \cdot C^* = \frac{a_C}{a_F + a_C + a_S} I, \quad
   p_S \cdot S^* = \frac{a_S}{a_F + a_C + a_S} I.    \]
So long as the consumer's income and the exponents in the utility function do not change, these solutions indicate that on a weekly basis, the consumer will spend identical \textit{amounts of money} on food, clothing, and shelter, regardless of prices.  Quantities are adjusted whenever prices change, but in a very peculiar way.  If food prices rise sharply, the quantity of food purchased is adjusted.  This we expect.  However, the quantities of clothing and shelter are not altered by a consumer with a Cobb-Douglas utility function!  This is \textit{not} how we tend to behave.  If food becomes more expensive and our budget becomes more strained than before, many of us will tend to buy clothes less frequently---decreasing the quantity of clothing purchased and the amount of money spent on clothing---even though the price of clothing has not changed. Similarly, many of us will tend to respond to sharply higher food prices by reducing the quantities of, and the amount of money spent on, those shelter-related items---we may push the thermostat lower in the winter and higher in the summer, reduce water usage, phone-calling, even disconnect the cable under really drastic circumstances---even though the prices of these shelter-related items have not changed.

Is there an obvious correspondence between the ordinary demand functions above and the results presented in \cite{bib:Morrill}?  Absolutely!  We could summarize the three functions in equation~(\ref{eq:my_F/c/s}) by using $X$ as an index:
\begin{equation}
\label{eq:demand_x}
     X^* = \frac{a_X}{a_F + a_C + a_S} \cdot \frac{I}{p_X} , 
                                       \; (X = F, C, S) . 
\end{equation}
Since the labor-leisure choice has already been made, the total amount of money available to spend on $X$ is $I$.  If we want one more unit of $X$, then we must give up an amount of money equal to its price.  The unit price of $X$ is $p_X$, so if we take the total amount of money, $I$, and divide by the unit price of $X$, then $I/p_X$ represents the largest feasible value that $X$ can take.  This quantity is multiplied by the share of $X$'s exponent relative to the sum of all of the exponents.

Now consider the optimal quantity of labor hours worked given in \cite{bib:Morrill} \textit{when the amount of non-labor income is zero}:
\begin{equation}
\label{eq:Morrill's_h}
        h^* = \frac{\alpha}{\alpha + \beta} \: T ,
\end{equation}
and use $L + h = T$ to write the consumer's optimal quantity of leisure:
\begin{equation}
\label{eq:Morrill's_L}
        L^* = T - h^* = \frac{\beta}{\alpha + \beta} \: T .
\end{equation}
(We convert hours-worked to hours of leisure because we always want to deal with ``goods,'' those things that make us happier when we have more of them. Although we all can enjoy our jobs, between labor and leisure, leisure is the ``good.'')  Does equation~(\ref{eq:Morrill's_L}) look like equation~(\ref{eq:demand_x})?  It certainly does!  In equation~(\ref{eq:Morrill's_L}), $T$ is the largest feasible value that $L$ can take.  This quantity is multiplied by the share of $L$'s exponent relative to the sum of all of the exponents, i.e., 
$\beta / (\alpha + \beta)$.

If you are not convinced, then let us look at this labor-leisure problem from a more economic perspective.

If the consumer has no non-labor income ($M = 0$ in \cite{bib:Morrill}), what is the largest feasible value that $C$ can take?  If the worker takes no leisure, $L = 0$, choosing to work during all of the $T$ hours available, then the worker's time worked is 
$h = T$ and the available income is $wT$; consequently, the consumer can have current consumption of $C = wT$.  

Now suppose that the worker/consumer chooses to have a single hour of leisure.  Since the sum of leisure hours and labor hours must be the total time available, $L + h = T$, $L = 1$ implies that $h = T - 1$.  How much current consumption can the consumer have while taking one hour of leisure?  Clearly, $C = wh = w(T - 1) = wT - w$.  Comparing the current consumption possible with $L = 1$ and $L = 0$, we find that current consumption is reduced by $w$, the wage rate.  As a result, economists say that the ``cost'' of an hour's leisure is the labor income forgone, namely the wage rate.  This can be interpreted as a economic formalization of the ``time is money'' phrase we hear so often.  

If time is money, then we can convert to money terms the $T$ hours available for allocation between labor and leisure.  Using the wage rate as the ``price'' or value of an hour's time, the worker/consumer has an amount of money $wT$ that s/he can spend on current consumption or leisure.  Thus, if no leisure is taken, all $wT$ dollars can be spent on current consumption, affording an amount of current consumption equal to $wT$.  If no current consumption is taken, all $wT$ dollars can be spent on leisure.  But an hour of leisure costs $w$, so the largest feasible value that $L$ can take is the total amount of money available, $wT$, divided by the price of one unit of $L$, namely $w$.  Therefore, the largest feasible value that $L$ can take is $wT/w = T$, which we already knew, but now we've worked this out using a ``price'' for an hour of leisure.

The upshot is still that $h^*$ (and $L^*$) are independent of $w$. To what, then, do we attribute this seemingly paradoxical result?  The Cobb-Douglas functional form!  To see that the Cobb-Douglas utility function is to blame, use the following utility function,
\begin{equation}
\label{eq:CES}
        U(C,L) = \left[ a C^{-\rho} + (1 - a)L^{-\rho} \right]^{-1/\rho} , \quad
                                                    0 < a < 1, \;\: \rho \geq -1 ,
\end{equation}
and follow the same procedure as in \cite{bib:Morrill}.  If one performs the necessary calculus, one obtains the following equivalent expressions for the optimal quantity of leisure:
\begin{align*}
L^* & = \frac{w \cdot T}
             {\left( \frac{\textstyle a}{\textstyle 1-a} \right)^{1/(\rho + 1)}                                                         \cdot w^{1/(\rho + 1)} + w} \;\;
      = \frac{(1 - a)^{1/(\rho + 1)} \cdot w \cdot T}
        {a^{1/(\rho + 1)} \cdot w^{1/(\rho + 1)} + (1 - a)^{1/(\rho + 1)} \cdot w} \\
    & = \frac{(1 - a)^{1/(\rho + 1)} \cdot T}
           {a^{1/(\rho + 1)} \cdot w^{-\rho/(\rho + 1)} + (1 - a)^{1/(\rho + 1)} } .
\end{align*}
Try as we might, $w$ cannot be eliminated from $L^*$.  This implies that $L^*$ depends on $w$, and so, therefore, will $h^*$ depend on the wage rate.  Thus, it is the Cobb-Douglas utility function that is to blame for the result in \cite{bib:Morrill} because the utility function in equation~(\ref{eq:CES}) is more general than the Cobb-Douglas function.  Indeed, a Cobb-Douglas utility function is a special case of the utility function given above; it is found as the limit of the above function as $\rho$ tends to zero:
\[ \lim_{\rho \rightarrow 0} 
      \left[ a C^{-\rho} + (1 - a)L^{-\rho} \right]^{-1/\rho} 
      = C^a L^{1-a} ; \]
see Problem \#703 in the \textit{Journal} \textbf{9}(2) (Spring~1990, p.~134).  Note also that in the limit as $\rho$ tends to zero, $L^*$ approaches $(1 - a) \cdot T$, which is independent of $w$, the result derived in \cite{bib:Morrill}.

\section*{Expunge the Cobb-Douglas?}
Economists as a group wish to explain the behavior we all observe in markets. Since most of us---as buyers of goods-and-services and sellers of labor services---do not behave in ways that can generally be explained by the Cobb-Douglas utility function, should we hit the textbooks and blot out all mentions of this particular function?  Of course not!  

The Cobb-Douglas utility function is useful for illustrating many economic principles in simple ways.  In a real sense, it is the best function on which beginning students can ``cut their analytical teeth.''  It is not a particularly realistic function to use, but in the early stages of study in any discipline, realism is frequently sacrificed, if temporarily, as a cost willingly paid to acquire intuition and understanding.  As we move progressively deeper in the study of any discipline in pursuit of realism and the desire to explain the world around us, we generally begin dropping the assumptions made for convenience and simplicity in the earliest stages of study.  To cite one obvious example, consider the study of ``mechanics'' in physics classes.  In the beginning we are allowed---even told---to ignore friction.  Is this realistic?  Of course not.  Indeed, each one of us has scraped elbows and knees and received burns from carpeting and gym floors; from an early age we have painful real-world experience with friction, and any theory of physics that ignores friction is destined to be imprecise.  But this is not wrong, as long as we are aware of the limitations of our assumptions.

\section*{Conclusion}
In \cite{bib:Morrill} the author asks several questions.  One of them, however, is this:  ``is the Cobb-Douglas model at fault?''  Clearly, the answer is a resounding ``yes!''  The Cobb-Douglas utility function is to the economic analysis of consumer behavior what the equilateral triangle is to geometry:  it is a ``special case'' in the extreme.  Any attempts to make general statements about consumer behavior based on results obtained from the Cobb-Douglas utility function would be about as useful as a flight plan for bringing a space shuttle back through the atmosphere without accounting for friction.  

Finally, can the Cobb-Douglas function's complicity come as a total surprise?  Certainly not, for between mathematician Charles~W.\ Cobb and economist Paul~H.\ Douglas, one of them was a politician (three-term U.S.~Senator)!

\begin{thebibliography}{9}
\bibitem{bib:Dudley} Dudley, Underwood, Editorial Comment, 
    \textit{Pi Mu Epsilon Journal} \textbf{10}(2) (Spring 1995), 130--133.
\bibitem{bib:Mitchell} Mitchell, Thomas, Discouraging ``Proof by Example,''
    mimeo, Department of Economics, Southern Illinois University at 
    Carbondale (1991).  This paper can be downloaded off the World Wide Web; 
    point a Web ``browser'' to: \\
    \mbox{ } \hfill 
    \texttt{http://econwpa.wustl.edu/eprints/ge/papers/9506/9506001.abs}. \hfill 
    \mbox{ } 
\bibitem{bib:Morrill} Morrill, John E., How Economists Use Mathematics
    To Show Why Some People Work So Much for So Little, \textit{Pi Mu
    Epsilon Journal} \textbf{10}(4) (Spring 1996), 269--272.
\end{thebibliography}

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\mbox{} \\
The author holds a bachelor's degree in mathematics and graduate degrees in economics; he is an associate professor of economics at Southern Illinois University at Carbondale, and a father to two daughters who cannot believe he reads the \textit{Pi Mu Epsilon Journal} ``for pleasure''!  Send electronic correspondence to \texttt{tmitch @@ siu.edu}. \\ \hrule
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