Suppose $f,g$ functions from $ A$ to $A$, f is injective and $g \circ f = f $ prove/disprove that $g$ is the identity function.

Question

I'm sure that I need to disprove this statement, but I spent alot of time trying to find two functions that fulfill the conditions but $g$ is not the identity function, any of my attempts shows that $g$ is actually the identity even I intuitive know it's not. What should I do?

Try it with a function f that is not surjective – Lasse Wulf Jul 04 '20 at 15:33 — Lasse Wulf, Jul 04 '20 at 15:33

score 1 · Answer 1 · answered Jul 04 '20 at 15:42

1

Your intuition is right. A solution could be, for instance, $A=\omega$, the set of the natural numbers, $f(n)=2n$ and $g(n)=n$ if $n$ is even and $g(n)=0$ if $n$ is odd.

There are many other example that work, the important thing is for $f$ not to be surjective. In particular, this implies that $A$ has to be infinite.

answered Jul 04 '20 at 15:42

Leo163

2,647

A link for infinite cardinal statement: https://math.stackexchange.com/a/63073/399263 – zwim Jul 04 '20 at 16:13

Suppose $f,g$ functions from $ A$ to $A$, f is injective and $g \circ f = f $ prove/disprove that $g$ is the identity function.

1 Answers1