Date: Oct 17, 2012 12:44 PM
Author: Paul A. Tanner III
Subject: Re: In favor of teaching "dot notation"
On Wed, Oct 17, 2012 at 9:57 AM, Joe Niederberger <firstname.lastname@example.org> wrote:
> Kirby says:
>>Lots of interesting musings and speculations, but I see nothing there to dissuade us from sharing more widely and effectively what 21st century math notations include
> Functional or "to one" relationships are pretty well understood today, no need to imbue them with mystical powers of action, or connotations of "belonging to" or "contained within". By the way, which is it? "Belong to"? or "contained in", or yet more? Is "dot" a superverb?
> Joe N
What do you mean?
The term "function" can be defined only with reference only to a set of ordered pairs in question, or it can be defined with reference to not just the set of ordered pairs in question but with reference to a superset of the set of all second elements in the set of ordered pairs or with reference to a superset of the set of all first elements in the set of ordered pairs or both.
Defining the term without reference to a codomain would be something like "a function is a set of ordered pairs such that no two of the first elements are equal".
But when a codomain is involved in its definition, the same set of ordered pairs can be a surjective function or a surjection depending on what set is defined to be the codomain - that is, depending on whether or not the range is a proper subset of the set defined to be the codomain.
And then we can have the same set of ordered pairs be called a partial function or a total function, this time depending on whether or not the set of first elements of the set of ordered pairs called the domain is a proper subset of the set defined to be what some call the superdomain or maximal domain of the partial function. (The HarperCollins Dictionary of Mathematics
calls it the maximal domain.)
If you are not familiar with partial or total functions, see this for an introduction:
(Here we see that some call the "larger" set in question the domain of a partial function while they call the set of first elements in question the domain of definition.)
So to sum up: Knowing whether a given function is is partial or total and whether it is surjective or not requires references to sets other than the sets of all first elements and second elements of the set of ordered pairs in question.