# Re: Graph

From: mAsterdam <mAsterdam_at_vrijdag.org>
Date: Tue, 15 Jan 2008 04:56:32 +0100
Message-ID: <478c2da0\$0\$85777\$e4fe514c_at_news.xs4all.nl>

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.

>
> Are you comparing visualisation with formalisation? You seem to be
> saying both are useful and important - and I agree.
>
> My point is only that the formal notion of a graph of a function (with
> no restriction to the reals) appears reasonably often in the
> mathematics literature.

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.

>>>>> 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).
>> ).

>
> What I mean is that it's such an insignificant thing, and not worth
> Keith's aggressive posts.

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.

>
> It appears we may have a different interpretation of what "is-a"
> means.

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
>> 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.

>
> But the cat's out of the bag! Are you suggesting for example that the
> "closed graph theorem" should be renamed the "closed plot theorem"?

No. You are.

--
What you see depends on where you stand.
Received on Tue Jan 15 2008 - 04:56:32 CET

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