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\begin{center}
{\mymagthreebf MATHEMATICS 300\ ---\  SPRING 2004}

\bigbreak

{\mymagtwoit Introduction to Mathematical Reasoning}\\

 {\mymagtwoit (Math 300)}

\bigbreak

{\mymagtwoit  H. J. Sussmann}

\bigbreak

\vspace{0.7cm}

{\Large{\bf  INSTRUCTOR'S NOTES, PART 4}}\\
{\large{\em (March 3, 2004)}}

\end{center}

\bigbreak


\begin{minipage}{12cm}
\baselineskip=9pt
{\tiny\tableofcontents}
\end{minipage}

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#2\ \ \hrulefill\ \hbox to 0.9cm{\hfill\ Page}\hbox to 0.6cm{\hfill #3}}

\section{Beware of mock turtles!}

Many students have repeatedly talked, in the homework, 
about ``the set $A\wedge B$'' (in a situation where 
$A$ and $B$ were two given sets), or ``the number $\sim x$''
(in a situation where $x$ was some given number, e.g., an integer
or a real number). 

This is something you should {\bf\em never} do. ``$\wedge$'' is a 
{\em binary logical connective}: it connects two {\em sentences}.
It does not connect things, such as numbers or sets. 
Similarly, 
``$\sim$'' is a 
{\em unary logical connective}: it applies to one {\em sentence}.
not to a thing such as a number or a set. There is no such thing as
``the set $A\wedge B$'' or ``the number $\sim x$''

I think the confusion arises from the following situations: students
often see sentences such as ``$x$ belongs to $A$ and $B$'',
or ``$\sim x=5$'', and somehow think that 
there exists a set called ``$A$ and $B$'' to which 
$x$ belongs, or a number ``$\sim x$'' which is equal to $5$. 
The cause of the confusion is {\bf\em incorrect grouping of words
or symbols.} 

The classical example of such incorrect grouping is Lewis Carroll's 
invention of a fictional character called ``The Mock Turtle,''
based on children's common misreading of the words
``mock turtle soup'' as ``(mock turtle) soup'', that
is, ``a soup made of mock turtle'', which would require 
the existence of an animal called ``mock turtle'' that this
soup would be made of.   The correct reading is,
of course, ``mock (turtle soup)'', that is, a turtle soup 
(or, better, a soup which is like turtle soup) but is really
an imitation turtle soup. (Actually, 
``mock turtle soup'' is  a soup made from a calf's head
to resemble turtle soup.)

Similarly, the correct reading of ``$\sim x=5$'' is
``$\sim(x=5)$'', that is, ``$x$ is not equal to $5$'',
or ``it's nottrue that $x=5$.'' (We don't write the parentheses
because they are not needed, since there is no other possible way 
to read ``$\sim x=5$''. You might think that 
it can be read as ``$(\sim x)=5$'', but this would not make any sense,
because ``$\sim x$'' is not anymore real than a mock turtle!


\section{Definitions: why they matter and how you should write them.}

We have already seen examples of why definitions are important.
In Problem 2 of Page 9 of Part 1 of these notes, 
we asked the question ``how many solutions does Problem 1
have?'' We reached the conclusion that we could not answer the question
because we did not know exactly what the precise meaning of 
``solution of Problem 1'' 

If we are asked ``is the number $23$ small?'', we have the same difficulty.
I have no idea how to answer this question, because I do not know
what ``small'' means. (Do you? If you think you do, please tell me!)
I know what ``smaller than'' means, but that's a totally different thing!

On the other hand, if we are asked ``is the number $18$ even?'',
then we can answer, because we have a precise definition of ``even'':

\medbreak
\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
{\Large 
\begin{center}
DEFINITION OF ``EVEN''
\end{center}

Let $n$ be an integer. We say that $n$ is {\bf\em even}
if there exists an integer $k$ such that $n=2k$.}
\end{minipage}}}
\end{center}

\medbreak\noindent We can also say this (but we don't have to) 
using symbolic language:
$$
\forall n\in\zz,(n\,\,{\rm is\,\,}{\bf even}\,\,
\Longleftrightarrow \exists k\in\zz,n=2.k)\,,
$$
or, if we have already somehow stipulated that our letter variables
take values in the set $\zz$ of all integers:
$$
\forall n,(n\,\,{\rm is\,\,}{\bf even}\,\,
\Longleftrightarrow \exists k,n=2.k)\,,
$$
To prove that $18$ is even we apply the definition.  Since the $n$ of the
definition is an {\bf\em arbitrary} integer, we can certainly take
$n=18$. Then we conclude that ``$18$ is even provided that there is an
integer $k$ such that $2k=18$.''  So now all we need is to prove that
``there is an integer $k$ such that $2k=18$.''  That is, we have to prove
that $\exists k\in\zz,2k=18$.  (Notice that here $k$ is a {\bf\em dummy
variable}, because it is {\bf\em bound variable}, 
occurring in a quantified statement under the scope of a quantifier.
We could equally well
have written this statement as $\exists x\in\zz,2x=18$, or $\exists
a\in\zz,2a=18$, or $\exists q\in\zz,2 q=18$, or
$\exists\alpha\in\zz,2\alpha=18$, or
$\exists\diamondsuit\in\zz,2\diamondsuit=18$, or
$\exists\clubsuit\in\zz,2\clubsuit=18$, or
$\exists\aleph\in\zz,2\aleph=18$, or
\mbox{$\exists something\in\zz,2 something=18$}, or 
$\exists gnu\in\zz,2 gnu=18$. The predicate 
``$\exists k,n=2.k$'' is a {\bf\em one variable predicate}---the variable
being $n$---because in order to decide whether 
``$\exists k,n=2.k$'' is true you need to ask me what $n$ is.
And the predicate 
``$\exists k,18=2.k$'' is a {\bf\em zero variables predicate}---i.e., a
statement---because in order to decide whether 
``$\exists k,18=2.k$'' is true you don't need to ask me anything,
because you can figure out whether the sentence is true 
or not all by yourself.)

Now recall our Rule $\exists_{in}$: 

\begin{center}\fbox{\fbox{\begin{minipage}{11cm}
{\Large\bf To prove that there exists a thing such that
something involving this thing 
happens, you show one (that is, you exhibit a witness).}

For example: to prove that there exists a town in New Jersey with a
population larger that 10,000, you say ``for instance, Piscataway
has more than 10,000 people.'' (Naturally, you could also have used
Trenton, or Princeton, or New Brunswick.)
\end{minipage}}}\end{center}

\noindent So, to prove that $\exists k\in\zz,2c=18$
you must exhibit an integer $k$ such that $2k=18$. The obvious choice (in
fact the only choice) is $9$. Since $2.9=18$, and $9$ is an integer, it is
true that $\exists k\in\zz,2k=18$.  So $18$ is even.

This definition takes care, in a completely unambiguous way, of
questions that students sometimes ask in class. For example, students
sometimes ask ``is $0$ even?'', or ``is $0$ considered even?'' My answer to
that question is ``if the definition of even implies that $0$ is even then
$0$ is even; if the definition of even implies that $0$ is not even then
$0$ is not even; to decide whether $0$ is or is not even, apply the
definition.''  So, you see, {\em having a precise definition of ``even''
enables you to settle the question whether something is even in cases when
you are in doubt.} (I should have said ``even in cases when you are in
doubt,'' but that would have looked bad, wouldn't it?)

Now consider the definition of ``prime.'' Is $3$ prime?
Is $6$ prime? Is $1$ prime?  Students often say vague, fuzzy things
such as ``a prime number is a number that has no factors.'' 
Now, {\em this purported definition is either wrong or 
so vague that you cannot work with
it.} Actually, every integer has ``factors'', since, for example, every
integer is divisible by itself and by $1$. So if one takes the 
definition literally, then no integer is prime, and the definition
is wrong, because this is obviously, not what you want!
So maybe you didn't quite mean that. So, what {\bf\em did}
you mean? Maybe when you said ``no factors'' you meant 
``no factors other than itself and $1$.'' But then, again,
that would not work, because every integer is also divisible 
by $-1$. So maybe you mean
``a prime number is a natural number that has no natural number
factors other than $1$ and itself.'' Now, this is a precise
definition, but turns out not to be a good one, because
according to this definition $1$ would be prime, and there are millions
of reasons why we do not want to count $1$ as prime. (For example: 
we want every natural number $>1$ to be uniquely expressible as
a prime or a product of primes. If $1$ was a prime, then we would have
$6=2.3$, but also $6=1.2.3$, and $6=1.1.1.1.1.1.1.1.1.2.3$, 
and this would violate the uniqueness of the factorization.) 

So, the usual ways of defining ``prime'' that students are used to 
are too vague to be useful. That's why we give a precise definition:

\medbreak
\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
{\Large 
\begin{center}
DEFINITION OF ``PRIME''
\end{center}

Let $n$ be an integer. We say that $n$ is {\bf\em prime}
if $n>1$ and there do not exist integers $p,q$  such that $n=p.q$
and $1<p<n$.}
\end{minipage}}}
\end{center}

\medbreak\noindent We can also say this (but we don't have to) 
using symbolic language:
$$
\forall n\in\zz,\Big(n\,\,{\rm is\,\,}{\bf prime}\,\,
\Longleftrightarrow(n>1\wedge\sim\exists p\in\zz,\exists q
\in\zz,(n=p.q\wedge 1<p<n))\Big)\,,
$$
or, if we have already somehow stipulated that our letter variables
take values in the set $\zz$ of all integers:
$$
\forall n,\Big(n\,\,{\rm is\,\,}{\bf prime}\,\,
\Longleftrightarrow(n>1\wedge\sim\exists p,\exists q,
(n=p.q\wedge 1<p<n))\Big)\,.
$$
This definition is truly useful, in the sense that you can work with it.
If I give you any integer, you can actually use the definition
and decide whethet the given integer is or is not prime. 

Similarly, if I am asked whether Star Trek's Mr. Spock has a brother,  then
I need a definition of ``brother'' that I can work with.  And, when I try
to write down such a definition, I am faced with the problem 
that the word ''brother,'' as is commonly used, is ambiguous, 
because sometimes we want to allow half brothers to count as brothers
and sometimes don't.  (And we also have to decide whether 
we are just dealing with human beings or we also count animals.
Here, for simplicity, I will just work with humans.)
So we have two possible ``formal'' definitions
of ``brother'': 

\medbreak
\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
{\Large 
\begin{center}
FIRST DEFINITION OF ``BROTHER''
\end{center}

Let $n,m$ be human beings. We say that $n$ is a {\bf\em brother}
of $m$ if $n$ is male, $n\neq m$, and the two 
parents of $n$ are also the parents of $m$. }
\end{minipage}}}
\end{center}

\medbreak\noindent We can also say this using symbolic language, 
if we write $\pp$ for ``the set of all human beings'': 
$$
\forall n\in\pp,\forall m\in\pp,\Big(n\,\,{\rm is\,\,a}\,\,{\bf brother}
\,\,{\rm of}\,\,m
\Longleftrightarrow((n\,\,{\rm is\,\,male}$$

\vspace{-7mm}
$$\wedge
n\neq m)\wedge\forall k\in\pp,
(k\,\,{\rm is\,\,a\,\,parent\,\,of\,\,}n\Longrightarrow
k\,\,{\rm is\,\,a\,\,parent\,\,of\,\,}m))\Big)
$$

\medbreak
\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
{\Large 
\begin{center}
SECOND DEFINITION OF ``BROTHER''
\end{center}

Let $n,m$ be human beings. We say that $n$ is a {\bf\em brother}
of $m$ if $n$ is male, $n\neq m$, and at least one of the two 
parents of $n$ is also a parent of $m$. }
\end{minipage}}}
\end{center}

\medbreak\noindent Again, we can athis using symbolic language:
$$
\forall n\in\pp,\forall m\in\pp,\Big(n\,\,{\rm is\,\,a}\,\,{\bf brother}
\,\,{\rm of}\,\,m
\Longleftrightarrow((n\,\,{\rm is\,\,male}$$

\vspace{-7mm}
$$\wedge
n\neq m)\wedge\exists k\in\pp,
(k\,\,{\rm is\,\,a\,\,parent\,\,of\,\,}n\wedge
k\,\,{\rm is\,\,a\,\,parent\,\,of\,\,}m))\Big)
$$
These  definitions are truly useful. And now we can decide
whether Mr. Spock has a brother: if we adopt the first definition,
he does not, but if we use the second definition he does.
(To be even more precise: everybody knows that 
Mr. Spock has an evil half-brother, so he does have a ``brother''
in the sense of the second definition.  But I am not 
aware of Mr. Spock having a brother in the sense of the first 
definition.)

Similarly, suppose I have to answer the question ``Are John
Kerry and Laura Bush married?'' If I try to understand what this means,
precisely, I run into the fact that the predicate ``married''
is ambiguous, because when you are talking about two people, 
``being married'' could mean ``being married to each other,''
or ``being married each one separately.'' So the ambiguity
here arises because, {\em there really are two predicates
``married'': one is the one-variable predicated ``begin
married to somebody'', and the other one is the two-variable predicate
``being married to each other.''} Let us call these predicates
``married$_1$'' and ``married$_2$''. Then we can give formal definitions:

\medbreak
\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
{\Large 
\begin{center}
DEFINITION OF ``MARRIED$_2$''
\end{center}

Let $n,m$ be human beings. We say that $n$ and $m$ are {\bf\em married$_2$}
if they have been legally declared spouses of each other.}
\end{minipage}}}
\end{center}

\medbreak
\begin{center}
\fbox{\fbox{\begin{minipage}{11cm}
{\Large 
\begin{center}
DEFINITION OF ``MARRIED$_1$''
\end{center}

Let $n$ be a human being. We say that $n$ is {\bf\em married$_1$}
if there exists $m$ such that $n$ and $m$ are married$_2$.} 
\end{minipage}}}
\end{center}

\subsection{Definitions and arguments}

From the previous examples, you can see that 


\medbreak\begin{center}\fbox{\fbox{\begin{minipage}{11cm}
Predicates have {\bf\em arguments}, so when you define
a predicate you have to begin by specifying what the arguments
are, and in particular it has to be clear from your definition
how many arguments there are. These arguments connect the
definition with the rest of the world. For example, the definitions of 
``even,'' ``prime,'' and ``married$_1$''
have {\bf one} argument each, so they enable you to plug in 
{\bf one} object and get a true-false conclusion. The definitions of
``divisible,'' ``brother,'' and ``married$_2$ ``
have  {\bf two} arguments. You plug in 
{\bf two} things to get a true-false conclusion.  
\end{minipage}}}\end{center}

\medbreak\noindent {\Large\bf When you write a definition you should
first of all figure out how many arguments will be involved, 
and then you should make sure that the definition begins 
by introducing these arguments.}

\begin{example} 
Let us figure out on our own, step by step, how to define
\mbox{``divisible.''} First of all, ``divisible'' is a word that talks about
two numbers.  (We do not talk about one number being ``divisible.'' We
talk about one number \mbox{being} \mbox{``divisible''} by another number.)  So if
we set out to write a definition of 
\mbox{``divisible''} we have to start by
introducing our two arguments: ``Let $a$ and $b$ be numbers.''
Moreover, the ``numbers'' involved, cannot just be any 
\mbox{numbers}. They
have to be {\em integers}.  So we really have to start with ``Let $a$ 
and $b$ be \mbox{integers}.'' Now that we know who we are talking
about (that is, we have properly introduced our two arguments) we are ready to
say what we want to say. We want to explain what it means
for $a$ to be divisible by $b$. So we write
``We say that $a$ is {\em divisible} by $b$ if (or `provided that')
XXXXXXXXXXX.''  Now what is missing is 
XXXXXXXXXXX. We already have the {\em structure} of our definition:

\medbreak
\begin{center}
Let $a$ and $b$ be integers. We say that $a$\\ is {\em divisible} by $b$
if  XXXXXXXXXXX.
\end{center}

\medbreak\noindent and we have to fill in the contents. 

So far, all  that mattered is that ``divisible'' is something
about two integers. Now we have to say what that ``something'' is.
In our case, the thing to put in the XXXXXXXXX slot
is ``$\exists c\in\zz,bc=a$,''
and we end up with 

\medbreak
\begin{center}
Let $a$ and $b$ be integers. We say that $a$ is\\ {\em divisible} by $b$
if  $\exists c\in\zz,bc=a$.
\end{center}
\end{example}

\begin{example} 
Let us figure out on our own, step by step, how to define
\mbox{``grandson''} for the benefit of someone who knows
what ``father,'' ``mother'' and ``male'' mean.
First of all, ``grandson'' is a word that talks about
two people.  (We do not talk about one person being ``a grandson.'' We
talk about one person being another person's \mbox{``grandson.''}) 
So if we set out to write a definition of 
\mbox{``grandson''} we have to start by
introducing our two arguments: ``Let $a$ and $b$ be persons.''
Now that we know who we are talking
about (that is, we have introduced our two arguments) we are ready to
say what we want to say. We want to explain what it means
for $a$ to be a grandson of $b$. So we write
``We say that $a$ is a {\em grandson} of $b$ if (or `provided that')
XXXXXXXXX.''  Now what is missing is 
XXXXXXXXX. We already have the {\em structure} of our definition:

\begin{center}
Let $a$ and $b$ be persons. We say that $a$ is a {\em grandson} of $b$\\
if  XXXXXXXX.
\end{center}

\noindent and we have to fill in the contents. (So far, all 
that mattered is that ``grandson'' is something
about two persons. Now we must say what that ``something'' is.)

In our case, the thing to put in 
the XXXXXX \mbox{slot} seems to be ``$b$ is a parent of 
someone who is a parent of $a$,'' that is,

\medbreak
``$\exists c,(c\,{\rm is\,\,a\,\,person\,\,and\,\,}c\,\,{\rm
is\,\,a\,\,parent\,\,of}\,\,a\,\,{\rm and\,\,}b\,\,{\rm
is\,\,a\,\,parent\,\,of\,\,}c$).'' 

\medbreak\noindent But a moment's thought shows that this is not enough.
We are trying to define {\em grandson}, not {\em grandchild}. So we have
to make sure that $a$ is male.  

The final result is 
\begin{center}
Let $a$ and $b$ be persons. We say that $a$ is a {\em grandson} of $b$ if\\
$a$ is male and $\exists c,(c\,{\rm is\,\,a\,\,person\,\,and\,\,}c\,\,{\rm
is\,\,a\,\,parent\,\,of}\,\,a\,\,{\rm and\,\,}b\,\,{\rm
is\,\,a\,\,parent\,\,of\,\,}c$). 
\end{center}
\end{example}

\subsection{Always highlight the definiendum}
When you write a definition, you are defining a particular word or phrase.
That word or phrase is called the {\em definiendum}. (This just means
``the thing being defined.'') {\bf The definiendum should always 
be highlighted.} In books, the authors do this by using Italics, or
Boldface.
(Look, for example, at any definitions you want in our textboook)
When you write your homework or your exams, or when  I write
on the blackboard, it's hard to do Italics, so we use 
underlining instead.

\subsection{More than two variables?}
As you probably have guessed by now, there are 
three-variable predicates, four-variable predicates, and so on.
For example, ``common divisor'' is a three-variable predicate:
let $a,b,c$ be integers; we say that $a$ is a {\bf\em comon divisor}
of $b$ and $c$ if $b$ is divisible by $a$ and $c$ is divisible by $a$.
(Note: I could have said 
``$b$ and $c$ are divisible by $a$,''  but then someone who believes
in the existence of the Mock Turtle might have drawn from this the wrong
conclusion that there is a number called ``$b$ and $c$.'')
\end{document}








