Re: Graph

David BL wrote:
> mAsterdam wrote:

>> David BL schreef:
>>> mAsterdam wrote:
>>>> David BL schreef:
>>>>> Keith H Duggar wrote:
>>>>>> David BL wrote:
>>>>>>> Keith H Duggar wrote:
>>>>>>>> David BL wrote:
>>>>>>>>> Marshall wrote:
>>>>>>>>>> An interesting note, by the way:
>>>>>>>>>> functions are relations ...
>>>>>>>>> Isn't it more precise to say that the graph of a
>>>>>>>>> function is a relation?
>>>>>>>> No, it isn't.
>>>>>>>> http://mathworld.wolfram.com/Function.html
>>>>>>> From mathworld a relation
>>>>>>>    http://mathworld.wolfram.com/Relation.html
>>>>>>> is defined as a subset of a cartesian product. If a
>>>>>>> function is a relation why do they define a graph of
>>>>>>> a function f as
>>>>>>>     { (x,f(x)) | x in domain of f },
>>>>>>> as described in
>>>>>>>    http://mathworld.wolfram.com/FunctionGraph.html
>>>>> ...  Wikipedia for example defines "graph of
>>>>> function" without any such restriction
>>>>>    http://en.wikipedia.org/wiki/Function_%28mathematics%29
>>>>>    http://en.wikipedia.org/wiki/Graph_of_a_function
>>>> Quoted from there:
>>>> "In mathematics, the graph of a function f is the collection of
>>>> all ordered pairs (x,f(x)). In particular, graph means the
>>>> graphical representation of this collection, in the form
>>>> of a curve or surface, together with axes, etc."
>>>> ISTM this is what I use the word 'plot' for.
>>>> Also from that page (at the start):
>>>> "For another use of the term "graph" in mathematics,
>>>> see graph theory".
>>>> In dutch 'grafiek' is a 'plot' (or 'chart'), and
>>>> 'graph' (another word) is a 'collection of edges and nodes'
>>>> - maybe it boils down to a homonym problem in english?
>>> I was exposed to the formal notion of a graph of a function in
>>> university when I studied functional analysis.   Check out the closed
>>> graph theorem
>>>    http://en.wikipedia.org/wiki/Closed_graph_theorem
>>> This is expressed on Banach spaces and has little to do with any
>>> visualisation.
>> Except for the foundational terms used; many have a strong
>> visual connotation: space, balls, vector (arrow),
>> topology (geographic maps), lineair (straight), orthogonal (right
>> angle), adjacency, distance, morphism (shapes),
>> let's not forget graph, and even plot.
>>
>> Visualizations do unnecessarily limit the dimensional
>> extend of conclusions to what we are able to
>> visualize (2D, 3D) so there is merit in separating definitions
>> from their visual connotations. This does not, however, make the
>> connotations go away or make them less useful for
>> learning the concepts.
>>
>> Is the distinction between two
>> (for other seehttp://en.wikipedia.org/wiki/Graph)
>> of the  meanings of graph (from graph theory vs.
>> graph of a function) purely a matter of different
>> visualisations? I do not think so.

I do notice that you are starting
to qualify 'graph' when you mean plot.
You say 'graph of a function'. I'll add that.

> For example it appears in the first few
> pages of my book "Elementary Classical Analysis" by Marsden and
> Hoffman.

I never read that book.

>>>>> This however doesn't change the fact that most authors define a
>>>>> (mathematical) relation as a set of ordered tuples, which means a
>>>>> function is not a relation (assuming, as most do, that a function has
>>>>> a defined domain and codomain).
>>>> ?
>>>> How does having a domain and a codomain stops a function from being a
>>>> kind of relation ? (David Cressey asked a similar question).
>>> Given the graph of a function you can determine its domain and range
>>> (also called image) but not its codomain.   Therefore a formal
>>> definition of a function tends to use the triple (D,C,G) where D is
>>> the domain, C is the codomain and G is the graph of the function.
>>> ISTM most authors only define a mathematical relation as a set of
>>> tuples (which can be compared to G).
>>> Saying that a function is not a relation is not terribly insightful
>>> and I'm sorry I said it!
>> Yet, you said it twice
>> (
>>  >>>>>>> Isn't it more precise to say that the graph of a
>>  >>>>>>> function is a relation?
>> and
>>  >>> This however doesn't change the fact that most authors define a
>>  >>> (mathematical) relation as a set of ordered tuples, which means a
>>  >>> function is not a relation (assuming, as most do, that a function
>>  >>> has a defined domain and codomain).
>> ).

Ah. Ok. I snipped both the unnecessary insult and your reaction as distracting.

>> You did not address my question.
>> I'll rephrase it as a statement:
>>
>> Having a domain and a codomain is relevant
>> to something being a function.
>> Having a domain and a codomain is irrelevant
>> to wether a function is a kind of relation or not.
>> You appear to see that differently. Please explain.

Now this is a kind of can of worm-like beings! I suspect it is more a matter of context: In the context of running programs, objects behave, and there is a concern for consistency in their behaviour: program correctnes, preferably provable. After the object dies, however, there still is data, which may later be used to incarnate similar objects, but also to build completely different objects.

For the data sec no such behavioral consistency concern applies.

A similar idea can be found at
http://alistair.cockburn.us/index.php/Constructive_deconstruction_of_subtyping, search for 'envelopes'.

> I am assuming that for a function to be a relation, a function
> is not permitted to introduce additional information (not available on
> the relation). Instead it is only allow to introduce constraints.
> You could compare this to Date's statement that it is wrong to say
> that a coloured rectangle is-a rectangle. This corresponds to
> thinking of a subtype as being a subset, and BTW is not the view
> generally held by most OO practitioners that assume is-a means LSP:-
>
> http://en.wikipedia.org/wiki/Liskov_substitution_principle

Which is, if I understood correctly, about behavioral consistency.

> In a pure mathematical setting, the "subtype = subset" view seems more
> appropriate, as for example when we say that a circle is-a ellipse.
>
>

>>  > It only has to do with what formalisms tend
>>  > to predominate in the literature.
>>
>> In which formalism is a function /not/ a kind of relation?
>>
>>>>> Furthermore, I was correct when I
>>>>> stated that a graph of a function is a relation, according to the more
>>>>> general definition of graph of function, as described in Wikipedia.
>>>> Is it more general?
>>>> Maybe so.
>>>> Anyway, what is wrong with using 'plot' for this, in order to
>>>> disambiguate - is there some meaning lost?
>>> To some people there is an established convention to say "graph of
>>> function" to formally refer to the set of ordered pairs, irrespective
>>> of any visual rendition.   Note that it is useful to have some
>>> terminology for this set, and it's not the first time that
>>> mathematical terms are overloaded.
>> No meaning lost by using plot. Define it as the set of ordered
>> pairs if you think it helps  - it's not that long. IMHO 'Graph'
>> as 'a collection of nodes and edges' is the more abstract
>> notion I would not want to demote.

No. You are.

--
What you see depends on where you stand.