Proving Validity of Function Compositions: A Comprehensive Guide

[Hobold]

Homework Statement

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) ;].

Homework Equations

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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

 

[Martin Rattigan]

You probably meant:

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

But it looks wrong at first sight.

 

[Hobold]

Yeah, that's exactly what I wrote

 

[Martin Rattigan]

Yeah, that's exactly what I wrote

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.

 

[Martin Rattigan]

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

Are both [itex]f\circ(g\circ h)[/itex] and [itex]h\circ(g\circ f)[/itex] defined? If so, are they equal?

 

[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!

 

[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).

 

[Martin Rattigan]

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