Function composition

1. Apr 30, 2010

Hobold

1. The problem statement, all variables and given/known data

Make [; f: A \rightarrow B ;], [; g: C \rightarrow D ;], [; h: E \rightarrow F ;] functions in which [; \text{Im} f \subseteq C;] and [; \text{Im} g \subseteq E;]. Show that [; f \circ ( g \circ h ) ;] and [; h \circ ( g \circ f ) ;] are valid if, and only if, [; f \circ ( g \circ h ) = h \circ ( g \circ f) ;].

2. Relevant equations

...

3. The attempt at a solution

Though the proof seems to be very trivial, I couldn't see very deeply.

I set the propositions necessary for the functions to exist, but I couldn't find a relation in the images, domains and codomains to make them equal.

Thanks

2. Apr 30, 2010

Martin Rattigan

You probably meant:

Make $f: A \rightarrow B$, $g: C \rightarrow D$, $h: E \rightarrow F$ functions in which $\text{Im} f \subseteq C$ and $\text{Im} g \subseteq E$. Show that $f \circ ( g \circ h )$ and $h \circ ( g \circ f )$ are valid if, and only if, $f \circ ( g \circ h ) = h \circ ( g \circ f)$.

But it looks wrong at first sight.

Last edited: Apr 30, 2010
3. Apr 30, 2010

Hobold

Yeah, that's exactly what I wrote

4. Apr 30, 2010

Martin Rattigan

Came out as:

Make [; f: A \rightarrow B ;], [; g: C \rightarrow D ;], [; h: E \rightarrow F ;] functions in which [; \text{Im} f \subseteq C;] and [; \text{Im} g \subseteq E;]. Show that [; f \circ ( g \circ h ) ;] and [; h \circ ( g \circ f ) ;] are valid if, and only if, [; f \circ ( g \circ h ) = h \circ ( g \circ f) ;].

on my screen. But there are some welly strange things happening with the Latex processing.

5. Apr 30, 2010

Martin Rattigan

Suppose $f:\mathbb{N}\rightarrow \mathbb{N}$ is $f:n\mapsto n+1$, $g=f$ and $h:\mathbb{N}\rightarrow \mathbb{N}$ is $h:n\mapsto max(n-2,0)$.

Are both $f\circ(g\circ h)$ and $h\circ(g\circ f)$ defined? If so, are they equal?

6. May 1, 2010

HallsofIvy

Martin, when you edit and it doesn't work correctly (and editing LaTex often gives that problem), try clicking on the "refresh" button. That often clears up the problem. Why it doesn't "refresh" automatically, I don't know!

7. May 1, 2010

Martin Rattigan

Thanks. With luck that should save me some work.

But in this instance it was Hobold's entry that was garbled and I hadn't edited it. In fact it still looks garbled on my screen (even after refresh).

8. May 1, 2010

Martin Rattigan

When I said, "But it looks wrong", I was referring to the content rather than the typesetting.