Hi John, and the other.

John motivates me to explain what is a function, "for a mathematician".

On 30 Jul 2009, at 17:53, John Mikes wrote:

> Hi, Bruno,
> let me skip the technical part

OK. But I remind you this current thread *is* technical.

> and jump on the following text.
> F u n c t i o n  as I believe is - for you - the y = f(x) form. For  
> me: the activity - shown when plotting on a coordinate system the  
> f(x) values of the Y-s to the values on the x-axle resulting in a  
> relation (curve). And here is my problem: who does the plotting? (Do  
> not say: YOU are, or Iam, that would add to the function concept the  
> homunculus to make it from a written format into a F U N C T I O N ).

But I have not yet say what is a function. I just mentioned that "they  
are everywhere" to open the appetite of the audience.

> F u n c t i o n  as I believe is - for you - the y = f(x) form.

You take a risk believing things - for me -.

Actually the "y = f(x)"  form will come later, with the goal of  
distinguishing clearly the key difference between a function and the  
many possible forms of a function.

Ah! but you force me to define what is a function right on (for a  
mathematician of course). Take it easy. You can skip to the sum-up  
line below.

OK, ready? I mean the others among those who pursue this mathematical  
shortcut toward the seventh step (the UD step, actually).

We have already seen functions. If you remember the bijection between  
A = {a,b,c,d,e,f,g} and B = {1,  2,  3,  4,  5,  6,  7}.

a  ----------  7
b  ----------  2
c  ----------  3
d  ----------  4
e  ----------  5
f  ----------  6
g  ----------  1

I said that the following set of couples

{(a,1), (b,2) (c,3) (d,4), (e,5), (f,6), (g,7)}

was a nice "set theoretical" representation of the bijection, and that  
the bijection is an example of function. We can give it a name, F, for  

F = {(a,1), (b,2) (c,3) (d,4), (e,5), (f,6), (g,7)}.

A function is a mathematical object, actually a set, which embodies an  
association between the elements of two sets. Here the two sets  
involved are A and B.
A is said to be the domain of F. B is said to be the range of F. And  
the function itself, F,  get a nice set theoretical  "form" of a set.  
The set of all the couples which determine or define the association.  
Here it is the set {(a,1), (b,2) (c,3) (d,4), (e,5), (f,6), (g,7)}.

Arbitrary set of couples will appear as very good way to describe  
relation, in general.

But for function a key condition, the functional condition, has to be  

       - If (a, b) belongs to F then if (a, c) belongs to F we have  
that b = c.   (the functional condition).

This means, that if F is a function from the set A to the set B, you  
cannot associate to one object of A, many objects of B.

For example the temperature in a place can be a function of time,  
because at each moment of time you will not associate two temperatures.
It is the key point for seeing that a function from A to B, describe a  
very general notion of dependency.

We will be interested in functions from N to N. (With N = {0, 1,  
2, ...}. Where examples abound.

Take the function which associates to each natural number its successor.

The function is (or is represented "fully") by the infinite set of  

{(0, 1), (1,2), (2,3), (3,4), (4,5), (5, 6), (6, 7), ...}

We will be interested in function having two arguments. Those will be  
the function from NXN to N. Example: take addition. This is a  
function, because when you add any numbers, 3 and 6, for example, 3+6,  
you don't expect two results. So the functional condition is  
respected. OK? So the function addition can be defined or represented  
by the set

{((0, 0), 0), ((0, 1) 1) ...  ((4, 8) 12) ... }

With the numbers, all the operations are functions. The same with the  

To sum up: a function is a set of couples, most of the time infinite,  
respecting the functional condition.

A good training consists in searching all functions between little sets:


1) how many functions and what are they, from the set {0, 1} to  
himself. What are the functions from {0, 1) to {0, 1}?


{(0,0), (1,0)}   the constant function which associates zero to any  
value of its argument.
{(0,1), (1,1)}   the constant function which associates one to any  
value of its argument.
{(0,0), (1,1)}  the identity function, which output its argument as  
{(0,1), (1,0)}, the NOT function, which associate 0 to 1, and 1 to 0.

There is four functions from {0, 1} to {0, 1}.

2) how many functions, and what are they, from the set cartesian  
product {0, 1} X {0, 1} to {0, 1}

Among them many are celebrities, you know. The AND, the OR, and many  
(how many?) others.

For a beginner in math, this is not at all an easy exercise. The real  
useful exercise is to try to understand the enunciation of the  
question. We will take the time needed.

3) A bit tricky perhaps: how many functions exist from { } to { } ?

----------------------------------------- SUM UP LINE  

So functions, once mathematical objects, are just set of couples,  
verifying a condition. We will be interested in the functions from N  
to N. Each such function is an infinite set of couples.

Some function have some form, or related expression. Not all though  
(as we will see), and we will have to study the relation between form  
and function. Many functions will lack a form, and this will not  
prevent them to play some role in the life of those who have a form.

John, we will see who plot which functions and why. I promise.  ;)



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