Re: Relationship between a function and its inverse

From: Adam (adam_at_bonkers.reg)
Date: 06/09/04 

Date: Wed, 09 Jun 2004 22:39:25 GMT

"Lee Rudolph" <lrudolph@panix.com> wrote in message
news:ca832d$gi7$1@panix2.panix.com...
> Suppose you use the formalization of "function" in which a function
> is identified with its graph: that is, a function f with domain X
> and codomain Y is identified with the subset of the cartesian product
> X x Y of all ordered pairs (x,f(x)). Now, as I'm sure you know,
> *any* subset R of X x Y can be interpreted as (and many people make
> this a definition) a "relation from X to Y". People who are using
> terms that way then say that the "inverse" of R is the relation
> R^{-1} from Y to X consisting of exactly those pairs (y,x) in Y x X
> such that (x,y) is in R. When X = Y, so that there is a natural
> "reflection across the diagonal" mapping X x X to itself by taking
> (x,y) to (y,x), then the relationship between R and R^{-1} is as
> you have described it above. Going back to the case of a function
> f from X to Y, although f^{-1} is always defined *as a relation* from
> Y to X, it will be a *function* from Y to X if and only if it is a
> bijection.

I can't say that I understand all of it at this point since I have not
learned about relations or bijections, since those are topics of the next
chapter. I'll start reading those topics. Hopefully then I will have an
image in my mind of how right and left inverses are related to functions,
since at this point I only know the definition.

Thanks for the explanation, Adam.

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