problem involving composite functions

Hi

Here is a problem I am trying.

Suppose and

and

I have to prove that f is a constant function.

I will give the outline of what I did.

Since

Now consider the set

g is a relation from A to A. Then I proved that g is also a function by proving

that

next , I proved that g is a constant function by proving that

and finally I used the given (1) for g , which implies that

To prove that f is a constant function, I have to prove

since , I can associate some such that

so Let

so the goal now is

Let x be arbitrary ,

since x is arbitrary

so f is a constant function..................

is the proof too detailed ? since this is from Velleman's "how to prove it" , I think,

author expects me to use all the logical machinery that I can use.

correct ? (Emo)

Re: problem involving composite functions

It's correct. Just as a matter of style though, it could be much more terse:

Since A is non-empty, let a be in A.

Let g = {<x a> | x is in A}.

g is a function from A into A.

Suppose x is in A.

So f(x) = f(g(x)) = f(a).

So f is a constant function.

Re: problem involving composite functions