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#161
 
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Re: Function -
01-20-2008
, 05:18 AM
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mAsterdam wrote: 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: [snipped changed entries] [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." 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. 3. Software A subroutine, procedure, or method. 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. notes: every operator is a function every function is a relation |
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It will be good to say something about some specifics that hold true for a function but doesn't hold for the relations in general. |
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1) The rule "f pairs each x with just one y" For example if one step on a scale then the scale will show just one number. The scale will not show two or three numbers in same time. We have here the law: weight = gravity x mass or y = gx. Because "f- machine" compute y, we can say that y is an extrinsic property of the person who step on the scale. Another example for above rule is a sequence of procedures: ----- > f1----- > f2 ------ > ... The sequence is very important for the structural programming and for some other sciences. |
'extrinsic'?
'sequence of procedures'?
'structural programming'?
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For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. |
Maybe it is a language thing.
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2) Construction of a relation (i) For example the following relation R ( identifier1, A1, A2) - where identifier1 get values from a procedure P, and D1, D2 are domains for A1, A2 - can't be constructed as the product of P x D1 x D2 because P is not set, it is the procedure. So to construct a relation we must have ready all sets. (ii) The pair primary key - foreign key, can find each other by matching. However m-n relationship need some third agent, usually a data entry person, who will pair every pair, it can be the millions of these pairs. On the other hand if we have a function, then for every x the function will compute the corresponding y. 3) Codomain The functions don't have so strict codomain as a relation. The codomain of function can be a superset of the range of the function. For example Bourbaki aproach to the function is based on the set of all functions from A to B. Here it is better to work with a codomain than with the range of the functions. This is longer text, |
I can't simply copy and paste it and say: this text improved (which I
could with the f-machine). Did you read the 'How to contribute' part?
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but the function is really basic thing. In fact I am surprise that people who are good at this field didn't write anything. |
Maybe it is good enough for the ones who are.
Bob Badour would prefer something different for
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every operator is a function if he'd care. |
--
What you see depends on where you stand.
#162
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This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
#163
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This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
#164
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This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
#165
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| |||
|
|
This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
#166
| |||
| |||
|
|
This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
#167
| |||
| |||
|
|
This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
#168
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|
|
This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
#169
| |||
| |||
|
|
This however, IMO complicates without need: 'extrinsic'? 'sequence of procedures'? 'structural programming'? |
and I separated some important characteristics of the functions in
three groups.
There is old definition of the function introduced by Leibniz, and
modern definition, defined by the group of mathematicians which wrote
under name Bourbaki. The Bourbaki definition became the first to
define function in terms of a set of ordered pairs. In Bourbaki
definition there is also part which define some additional conditions.
Now, there are discussants in this group, who insist that function is
just a kind of relation. Although the concept of relation is
important, insisting only on this concept can cause a misunderstanding
of the function.
In first part I wrote about importance of the rule: "f pairs each x
with just one y"
For example almost all laws in physics are in the form of a function,
satisfying this simple rule. Very important structures as the
sequences are implied by this rule (and not only by this rule). So I
don't think that this should go in the glossary, but it should be
clarified, because it is about basic things.
Quote:
|
For presenting function as "f-machine" we can use: Definition1 A function from A to B is a rule that assigns, to each member of set A, exactly one member of set B. Counter - example: In a theatre we can imagine a function from the set of visitors to the set of seats in the theatre. However we can't say what is the rule here. This is weak side of Definition1. We can use set-theoretic Definition2, formally it is OK for this example, but still we don't know what the rule here is. As far as I know there is no good definition for the algorithm; usually the algorithm is defined as the fixed set of rules. I do not understand what you are saying here. Maybe it is a language thing. |
This about "f-machine" should be supported by some background and it
is Definition1. If you note this definition doesn't have term
relation. Definition1 "smells" on a computation process. The counter-
example shows a weak side of Definition1. This can open a discussion
about the problems related to a definition of computable or algorithm,
what is not necessary for one glossary. Rather this is just one
interesting counter-example.
As conclusion, in my opinion you can add Bourbaki definition
(Definition2) now or later. So your glossary can cover "computable"
approach and also the set abstract approach. It will be OK and enough,
although there are some other approaches as the category theory for
example.
Vladimir Odrljin
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