\documentclass[12pt]{article}
\usepackage[Mickael]{ammaths}
\begin{document}
\module{Proof or spoof ?}{2}{02}{1 period}
%\prereq{}
\object{\begin{itemize}
\item Work on the concept of proof and discover some methods.
\end{itemize}}
\mater{\begin{itemize}
\item Titles and explanations for some methods of proof.
\end{itemize}}
\modpart{Matching game}{25 mins}
Students work in groups of 4 or 5. They are given the titles and explanations or examples for different kinds of proof,
all mixed up. They have to put the title back with the right explanation or example.
\modpart{Find the valid ones}{30 mins}
Each group has to find out which methods are valid and which ones are not.
\pagebreak
\moddocdis{Proof or spoof ?}{2}{02}{Explanations}
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
An offshoot of Proof by Induction, one may assume the result is true. Therefore it is true.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a ``proof without
words''. It is often used to prove the Pythagorean theorem.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Also called proof by example, it is the exhibition of a concrete example with a property to show that
something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by
constructing an explicit example.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Multiply both expressions by zero, e.g.,
\begin{eqnarray*}
1 &=& 2\\
1 \times 0 &=& 2 \times 0\\
0 &=& 0
\end{eqnarray*}
Since the final statement is true, so is the first.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
In this type of proof, the conclusion is established by logically combining the axioms, definitions, and earlier
theorems. For example, it can be used to establish that the sum of two even integers is always even:
\begin{quote}
Consider two even integers $x$ and $y$. Since they are even, they can be written as $x=2a$ and $y=2b$ respectively
for integers $a$ and $b$. Then the sum $x+y=2a+2b=2(a+b)$. From this it is clear $x+y$ has $2$ as a factor and
therefore is even, so the sum of any two even integers is even. This proof uses only definition of even integers and the
distribution law.
\end{quote}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
AN ARGUMENT MADE IN CAPITAL LETTERS IS CORRECT. THEREFORE, SIMPLY RESTATE THE PROPOSITION YOU ARE TRYING TO PROVE IN
CAPITAL LETTERS, AND IT WILL BE CORRECT!!!!!1 (USE TYPOS AND EXCLAMATION MARKS FOR ESPECIALLY DIFFICULT PROOFS)
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
By writing what seems to be an extensive proof and then smearing chocolate to stain the most crucial parts, the reader
will assume that the proof is correct so as not to appear to be a fool.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
In this important type of proof first a ``base case'' is proved, and then an ``inductive rule'' is used to prove a
(often infinite) series of other cases. Since the base case is true, the infinity of other cases must also be true,
even if all of them cannot be proved directly because of their infinite number.
Its principle states that: Let $\N=\{1,2,3,4,\ldots\}$ be the set of natural numbers and
$P(n)$ be a mathematical statement involving the natural number $n$ belonging to $\N$ such that
\begin{itemize}
\item[(i)] $P(1)$ is true, i.e., $P(n)$ is true for $n=1$ ;
\item[(ii)] $P(n+1)$ is true whenever $P(n)$ is true, i.e., $P(n)$ is true implies that $P(n+1)$ is true.
\end{itemize}
Then $P(n)$ is true for all natural numbers $n$.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Remember, something is not true when its proof has been verified, it is true as long as it has not been disproved. For
this reason, the best strategy is to limit as much as possible the number of people with the needed competence to
understand your proof.
Be sure to include very complex elements in your proof. Infinite numbers of dimensions, hypercomplex numbers,
indeterminate forms, graphs, references to very old books/movies/bands that almost nobody knows, quantum physics, modal
logic, and chess opening theory are to be included in the thesis. Make sentences in Latin, Ancient Greek, Sanskrit,
Ithkuil, and invent languages.
Again, the goal: nobody must understand, and this way, nobody can disprove you.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
In this type of proof, (also known as reductio ad absurdum, Latin for ``by reduction toward the absurd''), it is shown
that if some statement were so, a logical contradiction would occur, hence the statement must be not so. This method is
one of the most prevalent of mathematical proofs. A famous example shows that $\sqrt{2}$ is an irrational number:
\begin{quote}
Suppose that $\sqrt{2}$ is a rational number, so $\sqrt{2}=\frac{a}{b}$ where $a$ and $b$ are non-zero integers
with no common factor (definition of a rational number). Thus, $b\sqrt{2}=a$. Squaring both sides yields $2b^2=a^2$.
Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So
$a^2$ is even, which implies that $a$ must also be even. So we can write $a=2c$, where $c$ is also an integer.
Substitution into the original equation yields $2b^2=(2c)^2=4c^2$. Dividing both sides by 2 yields $b^2=2c^2$. But then,
by the same argument as before, 2 divides $b^2$, so b must be even. However, if $a$ and $b$ are both even, they share a
factor, namely 2. This contradicts our assumption, so we are forced to conclude that $\sqrt{2}$ is an irrational
number.
\end{quote}
%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
"I believe assertion A to hold, therefore it does. Q.E.D."
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
If enough people believe something to be true, then it must be so. For even more emphatic proof, one can use the similar
Proof by a Broad Consensus.
Either kind of proof can be combined with other types of proof (such as Proof by Repetition and Proof by Intimidation;
e.g., "A Broad Consensus of Scientists ...") when required.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
It establishes the conclusion ``if $p$ then $q$'' by proving the equivalent contrapositive statement
``if not $q$ then not $p$''.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
The proposition is true due to the lack of a counterexample. For when you know you are right and that you don't give a
shit about what others may think of you.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
In this type of proof, the conclusion is established by dividing it into a finite number of cases and proving each one
separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem
was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked
by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Make it easier on yourself by leaving it up to the reader. After all, if you can figure it out, surely they can.
Examples:
\begin{itemize}
\item "The reader may easily supply the details."
\item "The other 253 cases are analogous."
\item "The proof is left as an exercise for the reader."
\item "The proof is left as an exercise for the marker." (Guaranteed to work in an exam.)
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Since August is such a good time of year, then no-one will disagree with a proof published then, and therefore it is
true. Of course the converse is also true, i.e. January is shite, and all the logic in the world will not prove your
statement.
%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Reducing problems to diagrams with lots of arrows. This is related to proof by complexity.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
If there is a consensus on a topic. and you disagree, then you are right because people are stupid. See global warming
sceptics, creationist, tobacco companies, etc., for application of this proof.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Be sure to provide some distraction while you go on with your proof, e.g., some third-party announces, a fire alarm (a
fake one would do, too) or the end of the universe. You could also exclaim, "Look! A distraction!", meanwhile pointing
towards the nearest brick wall. Be sure to wipe the blackboard before the distraction is presumably over so you have the
whole board for your final conclusion.
Don't be intimidated if the distraction takes longer than planned and simply head over to the next proof.
An example is given below.
\begin{enumerate}
\item Look behind you!
\item ... and proves the existence of an answer for 2 + 2.
\item Look! A three-headed monkey over there!
\item ... leaves 5 as the only result of 2 + 2.
\item Therefore 2 + 2 = 5. Q.E.D.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
Suppose $P(n)$ is a statement.
\begin{enumerate}
\item Prove true for $P(1)$.
\item Prove true for $P(2)$.
\item Prove true for $P(3)$.
\item Therefore $P(n)$ is true for all $n$.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
If Jack Bauer says something is true, then it is. No ifs, ands, or buts about it. End of discussion.
%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
If you say something is true enough times, then it is true. If you say something is true enough times, then it is true.
If you say something is true enough times, then it is true. If you say something is true enough times, then it is true.
If you say something is true enough times, then it is true. If you say something is true enough times, then it is true.
Exactly how many times one needs to repeat the statement for it to be true is debated widely in academic circles.
Generally, the point is reached when those around die through boredom.
eg. let $A=B$ since $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ then
$A=B$.
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{tabularx}{\textwidth}{|c|X|c|c|}
\hline
\bf \textsf{Type of proof} & \rule{0pt}{15pt} & Valid & Not valid\\[5pt]
\hline
\end{tabularx}
\end{center}
If you prove your claim for one case, and make sure to restrict yourself to this one, you thus avoid any case that could
compromise you. You can hope that people won't notice the omission.
Example: Prove the four-color theorem. Take a map of only one region. Only 1 color is needed to color it and 1 is less
than 4. End of the proof.
If someone questions the completeness of the proof, others methods of proofs can be used.
\pagebreak
\moddocdis{Proof or spoof ?}{2}{02}{Types of proof}
\begin{center}
\begin{minipage}{12cm}\Large
\begin{itemize}
\itemb Visual proof
\itemb Proof by Multiplicative Identity
\itemb Proof by Restriction
\itemb Proof by Assumption
\itemb Proof by Belief
\itemb Proof by Construction
\itemb Proof by Cases
\itemb Proof by Complexity
\itemb Proof by (a Broad) Consensus
\itemb Proof by August
\itemb Proof by Default
\itemb Proof by Transposition
\itemb Proof by Delegation
\itemb Direct proof
\itemb Proof by Chocolate
\itemb Proof by Dissent
\itemb Proof by Distraction
\itemb Proof by Exhaustion
\itemb Proof by Engineer's Induction
\itemb Proof by Diagram
\itemb Proof by Mathematical Induction
\itemb Proof by Jack Bauer
\itemb Proof by Repetition
\itemb Proof by contradiction
\end{itemize}
\end{minipage}
\end{center}
\pagebreak
\moddoc{Methods of proof : titles and explanations}
\sectionblack{Direct proof}
In this type of proof, the conclusion is established by logically combining the axioms, definitions, and earlier
theorems. For example, it can be used to establish that the sum of two even integers is always even:
\begin{quote}
Consider two even integers $x$ and $y$. Since they are even, they can be written as $x=2a$ and $y=2b$ respectively
for integers $a$ and $b$. Then the sum $x+y=2a+2b=2(a+b)$. From this it is clear $x+y$ has $2$ as a factor and
therefore is even, so the sum of any two even integers is even. This proof uses only definition of even integers and the
distribution law.
\end{quote}
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by mathematical induction}
In this important type of proof first a ``base case'' is proved, and then an ``inductive rule'' is used to prove a
(often infinite) series of other cases. Since the base case is true, the infinity of other cases must also be true,
even if all of them cannot be proved directly because of their infinite number.
Its principle states that: Let $\N=\{1,2,3,4,\ldots\}$ be the set of natural numbers and
$P(n)$ be a mathematical statement involving the natural number $n$ belonging to $\N$ such that
\begin{itemize}
\item[(i)] $P(1)$ is true, i.e., $P(n)$ is true for $n=1$ ;
\item[(ii)] $P(n+1)$ is true whenever $P(n)$ is true, i.e., $P(n)$ is true implies that $P(n+1)$ is true.
\end{itemize}
Then $P(n)$ is true for all natural numbers $n$.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by transposition}
It establishes the conclusion ``if $p$ then $q$'' by proving the equivalent contrapositive statement
``if not $q$ then not $p$''.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by contradiction}
In this type of proof, (also known as reductio ad absurdum, Latin for ``by reduction toward the absurd''), it is shown
that if some statement were so, a logical contradiction would occur, hence the statement must be not so. This method is
one of the most prevalent of mathematical proofs. A famous example shows that $\sqrt{2}$ is an irrational number:
\begin{quote}
Suppose that $\sqrt{2}$ is a rational number, so $\sqrt{2}=\frac{a}{b}$ where $a$ and $b$ are non-zero integers
with no common factor (definition of a rational number). Thus, $b\sqrt{2}=a$. Squaring both sides yields $2b^2=a^2$.
Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So
$a^2$ is even, which implies that $a$ must also be even. So we can write $a=2c$, where $c$ is also an integer.
Substitution into the original equation yields $2b^2=(2c)^2=4c^2$. Dividing both sides by 2 yields $b^2=2c^2$. But then,
by the same argument as before, 2 divides $b^2$, so b must be even. However, if $a$ and $b$ are both even, they share a
factor, namely 2. This contradicts our assumption, so we are forced to conclude that $\sqrt{2}$ is an irrational
number.
\end{quote}
%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak
\sectionblack{Proof by construction}
Also called proof by example, it is the exhibition of a concrete example with a property to show that
something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by
constructing an explicit example.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by exhaustion}
In this type of proof, the conclusion is established by dividing it into a finite number of cases and proving each one
separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem
was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked
by a computer program, not by hand. The shortest known proof of the four colour theorem today still has over 600 cases.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Visual proof}
Although not a formal proof, a visual demonstration of a mathematical theorem is sometimes called a ``proof without
words''. It is often used to prove the Pythagorean theorem.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Multiplicative Identity}
Multiply both expressions by zero, e.g., let's prove that 1
$1=2$ :
\begin{eqnarray*}
1 &=& 2\\
1 \times 0 &=& 2 \times 0\\
0 &=& 0
\end{eqnarray*}
Since the final statement is true, so is the first.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by August}
Since August is such a good time of year, then no-one will disagree with a proof published then, and therefore it is
true. Of course the converse is also true, i.e. January is shite, and all the logic in the world will not prove your
statement.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Assumption}
An offshoot of Proof by Induction, one may assume the result is true. Therefore it is true.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Belief}
"I believe assertion A to hold, therefore it does. Q.E.D."
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Cases}
AN ARGUMENT MADE IN CAPITAL LETTERS IS CORRECT. THEREFORE, SIMPLY RESTATE THE PROPOSITION YOU ARE TRYING TO PROVE IN
CAPITAL LETTERS, AND IT WILL BE CORRECT!!!!!1 (USE TYPOS AND EXCLAMATION MARKS FOR ESPECIALLY DIFFICULT PROOFS)
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Chocolate}
By writing what seems to be an extensive proof and then smearing chocolate to stain the most crucial parts, the reader
will assume that the proof is correct so as not to appear to be a fool.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Complexity}
Remember, something is not true when its proof has been verified, it is true as long as it has not been disproved. For
this reason, the best strategy is to limit as much as possible the number of people with the needed competence to
understand your proof.
Be sure to include very complex elements in your proof. Infinite numbers of dimensions, hypercomplex numbers,
indeterminate forms, graphs, references to very old books/movies/bands that almost nobody knows, quantum physics, modal
logic, and chess opening theory are to be included in the thesis. Make sentences in Latin, Ancient Greek, Sanskrit,
Ithkuil, and invent languages.
Again, the goal: nobody must understand, and this way, nobody can disprove you.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by (a Broad) Consensus}
If enough people believe something to be true, then it must be so. For even more emphatic proof, one can use the similar
Proof by a Broad Consensus.
Either kind of proof can be combined with other types of proof (such as Proof by Repetition and Proof by Intimidation;
e.g., "A Broad Consensus of Scientists ...") when required.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Default}
The proposition is true due to the lack of a counterexample. For when you know you are right and that you don't give a
shit about what others may think of you.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Delegation}
Make it easier on yourself by leaving it up to the reader. After all, if you can figure it out, surely they can.
Examples:
\begin{itemize}
\item "The reader may easily supply the details."
\item "The other 253 cases are analogous."
\item "The proof is left as an exercise for the reader."
\item "The proof is left as an exercise for the marker." (Guaranteed to work in an exam.)
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Diagram}
Reducing problems to diagrams with lots of arrows. This is related to proof by complexity.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Dissent}
If there is a consensus on a topic. and you disagree, then you are right because people are stupid. See global warming
sceptics, creationist, tobacco companies, etc., for application of this proof.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Distraction}
Be sure to provide some distraction while you go on with your proof, e.g., some third-party announces, a fire alarm (a
fake one would do, too) or the end of the universe. You could also exclaim, "Look! A distraction!", meanwhile pointing
towards the nearest brick wall. Be sure to wipe the blackboard before the distraction is presumably over so you have the
whole board for your final conclusion.
Don't be intimidated if the distraction takes longer than planned and simply head over to the next proof.
An example is given below.
\begin{enumerate}
\item Look behind you!
\item ... and proves the existence of an answer for 2 + 2.
\item Look! A three-headed monkey over there!
\item ... leaves 5 as the only result of 2 + 2.
\item Therefore 2 + 2 = 5. Q.E.D.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Engineer's Induction}
Suppose $P(n)$ is a statement.
\begin{enumerate}
\item Prove true for $P(1)$.
\item Prove true for $P(2)$.
\item Prove true for $P(3)$.
\item Therefore $P(n)$ is true for all $n$.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Jack Bauer}
If Jack Bauer says something is true, then it is. No ifs, ands, or buts about it. End of discussion.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Repetition}
If you say something is true enough times, then it is true. If you say something is true enough times, then it is true.
If you say something is true enough times, then it is true. If you say something is true enough times, then it is true.
If you say something is true enough times, then it is true. If you say something is true enough times, then it is true.
Exactly how many times one needs to repeat the statement for it to be true is debated widely in academic circles.
Generally, the point is reached when those around die through boredom.
eg. let $A=B$ since $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ and $A=B$ then
$A=B$.
%%%%%%%%%%%%%%%%%%%%%%%%
\sectionblack{Proof by Restriction}
If you prove your claim for one case, and make sure to restrict yourself to this one, you thus avoid any case that could
compromise you. You can hope that people won't notice the omission.
Example: Prove the four-color theorem. Take a map of only one region. Only 1 color is needed to color it and 1 is less
than 4. End of the proof.
If someone questions the completeness of the proof, others methods of proofs can be used.
\end{document}