What are identities in elementary mathematics?

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What are identities in elementary mathematics? Sergei Akbarov 10/22/10 2:00 PM
Dear colleagues,

Who can explain me what people in Calculus (or, I do not know, maybe I
should say "in elementary mathematics") mean by equality of elementary
functions? I mean, everybody knows, of course, that equality of functions

f(x)=g(x)

on, say, an interval I means that they coinside in each point x\in I.

This is the definition for all, not necessarily elementary functions. But
one can expect that elementary functions (I mean, x^a, a^x, sin, cos, etc.)
can be defined independently in purely algebraic way, say, like an algebra,
generated by a list of identities. Of course, the definition must be such
that all the other identities could be deduced from this prearrannged list
as corollaries, and this will be equivalent to the definition of this
algebra as a subalgebra of all functions on an interval I.

People in computer algebra are discussing different ways to teach computer
to recognize identities in elementary mathematics, see, e.g the book "A=B"

http://www.math.upenn.edu/~wilf/AeqB.html

and the identities they consider are "analytical identities", i.e. to study
them we should consider our functional algebra as a subalgebra of the
algebra of all functions on I (or of all continuous, smooth, analytical,
etc., functions on I). But perhaps there is a way to separate elementary
functions from analysis? I mean to axiomatize them in such a way that the
question of whether f(x)=g(x) is true or not will be just a question,
whether this identity can be deduced in algebraic way from the axioms of the
theory?

Were there any investigations in this field? Does such approach indeed has a
chance to exist, or, maybe there are some negative results?

I would greatly appreciate any references, suggestions, etc.

Sergei Akbarov


Re: What are identities in elementary mathematics? Norbert Marrek 10/23/10 9:00 AM
Am 22.10.2010 23:00, schrieb Sergei Akbarov:

In Z/2Z the polynomial functions f(X)=X+1 and g(X)=X^2+1
have the save values f(9)=g(0)=1 and f(1)=g(1)=0.
Would you consider X+1 the same as X^2+1?

Aloha
        Norbert

Re: What are identities in elementary mathematics? tc...@lsa.umich.edu 10/23/10 9:00 AM
In article <i9su0g$m6b$1...@news.acm.uiuc.edu>,

Sergei Akbarov <sergei....@gmail.com> wrote:
>But perhaps there is a way to separate elementary functions from
>analysis? I mean to axiomatize them in such a way that the question of
>whether f(x)=g(x) is true or not will be just a question, whether this
>identity can be deduced in algebraic way from the axioms of the theory?

I think the top two Google Scholar hits on "recognizing zero" should
point you in the right direction.
--
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences

Re: What are identities in elementary mathematics? David Hobby 10/24/10 2:00 PM
On Oct 23, 12:00�pm, "tc...@lsa.umich.edu" <tc...@lsa.umich.edu>
wrote:
> In article <i9su0g$m6...@news.acm.uiuc.edu>,

>
> Sergei Akbarov <sergei.akba...@gmail.com> wrote:
> >But perhaps there is a way to separate elementary functions from
> >analysis? I mean to axiomatize them in such a way that the question of
> >whether f(x)=g(x) is true or not will be just a question, whether this
> >identity can be deduced in algebraic way from the axioms of the theory?
>
> I think the top two Google Scholar hits on "recognizing zero" should
> point you in the right direction.
> --
> Tim Chow � � � tchow-at-alum-dot-mit-dot-edu
...
As an aside, I'd expect axiomatizing the theory of elementary
functions
to be tricky.  Compare this to Tarski's High School Algebra Problem,
which considered a small subset of all elementary functions and was
still quite difficult.
http://en.wikipedia.org/wiki/Tarski%27s_high_school_algebra_problem

---David Hobby

Re: What are identities in elementary mathematics? Dan Luecking 10/25/10 5:00 PM

I certainly would, because surely x^2 = x is one of the
identities one would include as an axiom for elementary
functions over Z/2Z.

Similarly, I would consider e^x e^y the same as e^{x+y}
over the reals (thought not over the ring of n x n
matrices).


Dan
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Re: What are identities in elementary mathematics? mjc 10/28/10 1:00 AM
>
> I certainly would, because surely x^2 = x is one of the
> identities one would include as an axiom for elementary
> functions over Z/2Z.
>
> Similarly, I would consider e^x e^y the same as e^{x+y}
> over the reals (thought not over the ring of n x n
> matrices).
>
> Dan

I would consider both of these to be theorems, not axioms (the first
from 1+1=0, the second from the definition of e^x and properties of
exponentiation and limits).

Re: What are identities in elementary mathematics? Ilya Zakharevich 10/28/10 12:00 PM
On 2010-10-28, mjc <mjc...@acm.org> wrote:
>>
>> I certainly would, because surely x^2 = x is one of the
>> identities one would include as an axiom for elementary
>> functions over Z/2Z.
>>
>> Similarly, I would consider e^x e^y the same as e^{x+y}
>> over the reals (thought not over the ring of n x n
>> matrices).

> I would consider both of these to be theorems, not axioms (the first
> from 1+1=0,

I doubt it.  In F_4, 1+1=0, but x^2 is not equal to x.

> the second from the definition of e^x and properties of
> exponentiation and limits).

I do not know what is a "limit".  And I wonder which "property of
exponentiona" would imply "e^x e^y is the same as e^{x+y}"...

Ilya

  [P.S.  Is it my imagination, or had the quality of moderation
         slipped a bit during the last year?]

Re: What are identities in elementary mathematics? Dan Luecking 10/29/10 4:30 AM

I was referring to the original poster's discussion: axioms
for a system used to determine equality of elementary
functions in a purely algebraic way. This system needs a
list of rules or identities for transforming one function
into another. These would be the _axioms_ of such a system,
and they could certainly be required to depend on what the
variables represent.

For example, the algebra of elementary functions (over C)
is defined to be the smallest set of formulas that contains
all constants, z and e^z, and is closed under the usual
arithmetic operations plus also functional composition
and inversion. The usual field axioms would have to be part
of the set of rules (or axioms), and some of the laws of
exponents (since not all follow algebraicly from the field
axioms).

Over C, sin z can be defined in terms of e^z, and various
trig identities follow from identities for e^z. But over R,
one would have to add at least sin x to the starting list.
And some more identities for things like sin(x+y).

Finally, if our elementary functions are going to be
operating over Z/2Z, we need identities or axioms
appropriate for that system. Surely we would want our
system to include x^2 = x as this is both useful and,
in the form (x-1)x = 0, concisely expresses the single
defining property of Z/2Z among fields: every element is
either 0 or 1. If we did not include it we would have to
include something else that expresses the difference
between Z/2Z and R or C.

My main point was that the set of identities (or axioms)
differs depending on what the variables in our functions
represent. So x^2 is the same as x in Z/2Z (and not in R)
because some axioms make it so. In my previous post, I did
not say that the e^{x+y} = e^x e^y was an axiom, but that
it is an identity valid for real variables and not for
functions of matrix variables. (Still, it doesn't follow
from the field axioms unless x and y are rational, so
_some_ axiom would still have to be added, and the rules
of the game were that it has to be algebraic.)


 

Dan
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