# Graph (was: what are keys and surrogates?)

From: mAsterdam <mAsterdam_at_vrijdag.org>
Date: Mon, 14 Jan 2008 11:09:32 +0100
Message-ID: <478b3386\$0\$85779\$e4fe514c_at_news.xs4all.nl>

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 see http://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.

```>>> 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).
```
).

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.

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.

```--
What you see depends on where you stand.
```
Received on Mon Jan 14 2008 - 11:09:32 CET

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