Function

#41

On Jan 15, 6:25 am, mAsterdam <mAster... (AT) vrijdag (DOT) org> wrote:

Quote:

Jan Hidders wrote:

[snip]

(cdt glossary

[Function]
...
Math
A binary mathematical relation with at most
one b for each a in (a,b).

This "at most one b for each a in (a,b)" makes me cringe!

Irritation about the status quo is a starting point
to many improvements. I am sure the "Software" subentry (you
snipped it) makes functional programming adepts curl
their toes as well - I'll keep it until somebody provides
a better text.

Moreover, it
seems to describe partial functions, which is not what is usually
understood under "function". I would make that:

"A binary mathematical relation over two sets D and C that associates
with each element in D exactly one element in C."

I like it.
I'll post a proposal for replacement in my answer to Vladimir.
To the native english speakers: is 'that' correct in Jan's sentence?

Yes.

There are different kinds of things that variously get called
functions:

total functions, partial functions, multifunctions, aggregate
functions

Often "function" by itself means "total function" but sometimes it
doesn't.

It seems the difference between a total function and a partial
function is
just in what we want to call the domain. Division over the domain
(integer, integer) is partial; division over the domain (integer,
nonzero integer)
is total. What's up with that?

Some things out there produce more than one result, or a stream of
results. Sometimes these are called multifunctions and sometimes
generators.

Then we have things like sum() avg() etc. Aggregate functions.

Marshall

#42

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#43

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#44

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#45

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#46

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#47

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#48

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#49

On 2008-01-15, Marshall wrote:

Quote:

Often "function" by itself means "total function" but sometimes it
doesn't.

The standard reading of "function" implies "total". The only time
somebody qualifies it explicitly is when the context includes all these
"non-function functions" as well.

Quote:

It seems the difference between a total function and a partial
function is just in what we want to call the domain.

Yes. The concept of partial functions only exists because in the context
of relative mathematical properties, natural extensions and restrictions
relate partial functions defined on supersets to actual functions on
subsets. And of course because functionality is quite a useful and
restrictive property of a general relation as well.

Quote:

Division over the domain (integer, integer) is partial; division over
the domain (integer, nonzero integer) is total.

....which is an example of restriction of course.

Really, the weird part about this thread, to me, is how much time is
being spent on how various people construct relations, functions and the
like. In today's math it's much more common to go with the axiomatic
method and simply talk about the properties any such constructs possess.
Under that sort of treatment, most of the fuzziness goes away because
you can show that the various constructive versions are isomorphic to
each other; from the viewpoint of behavior, properties and logic, they
are all just models of the same basic mathematical intuition. Such a
viewpoint saves you a whole lot of quibbling.
--
Sampo Syreeni, aka decoy - mailto:decoy (AT) iki (DOT) fi, tel:+358-50-5756111
student/math+cs/helsinki university, http://www.iki.fi/~decoy/front
openpgp: 050985C2/025E D175 ABE5 027C 9494 EEB0 E090 8BA9 0509 85C2

#50

On 2008-01-15 09:26:01 -0500, mAsterdam <mAsterdam (AT) vrijdag (DOT) org> said:

Quote:

vldm10 wrote:
I think it will be good to have two definitions for the functions in
your glossary.
Definition1 A function from A to B is a rule that assigns, to each
member of set A, exactly one member of set B.

And second definition is similar to Jan's suggestion, but slightly
changed:
Definition2
A function from A to B is a relation between A and B that associates
each element of A with exactly one element of B.

First definition says that a function do something. You can call it
intutive definition of a function. Here the function in fact is a
procedure as you mentioned.
Second definition is set theoretic.

Another difference I see with Jan's is a sense of direction.

How about this:
cdt glossary proposal:

[Codomain]
See function, math context.

[Domain]
1. Given a relation R, a domain is a set Sn such
that for each tuple (A1, A2, ...An, ...Am) in R,
An is an element of Sn.

This is not good enough. It is possible that a value exists in the
domain Sn yet the relation has no corresponding tuple which holds that
value for An.

Quote:

2. A domain is a set of values: for example
"integers between 0 and 255",
"character strings less than 10 characters long",
"dates".
Sometimes used synonymously with type.

This seems right. A domain is just a set of values. In relational
algebra, this set is required to be non-empty since attributes are
non-null.

Quote:

3. Domain of a function. See function, math context.

On the other hand, mathematics does not require a domain to be non-empty.

Quote:

[Function]
For now we have to live with different meanings
of _function_ when talking about databases:
"The function of this function is to get the tuples from B
that are functionally dependant on A."

No, there is always just one meaning of function in database.

Quote:

Three different contexts, but just about the same meaning:

1. General
A purpose or use.

2. Math
A binary mathematical relation over two sets D and C that associates
with each element in D exactly one element in C.
Set D is called the domain of the function, C its codomain.

Essentially correct, although to be rigorous you need to define how
such binary relation can define the meaning of "associating each
element in D exactly one element in C."

Not all binary relation has this property.

Quote:

3. Software
A subroutine, procedure, or method.

Yea. It's really an abused use of the term in software design.
Subroutines in software has no clear domain since same input arguments
can product different outputs.

Quote:

In both the math and software context, there is a sense of
direction from domain (input) to codomain (output).
For most purposes, this intuitive picture is good enough:

|------------|
--- x ---- >| f-machine |------ f(x) -----
|------------|

Where x is input in the "f-machine" and f(x) is output.

Fair.

Quote:

notes:
every operator is a function
every function is a relation

Yes.

--

-kira