Proving f:A->A is One-to-One & Onto

[moo5003]

Homework Statement

Assume that A is finite and f: A->A. If f is onto then f is one-to-one

2. The attempt at a solution

f[A] = A
Card(A) = n a natural number.

I assume I need to use the axiom of choice since this equivalence only holds if you use it (correct?).

My proof so far is supposing that its not injective, then there would exist a_1, a_2, a_3..., a_k such that f(a_1)=f(a_2)...=f(a_k) = C in A.

Thus, f[A\{a_2, a_3, a_4.., a_k}] = A

I want to somehow keep reducing this until f is known to be injective and then show that there is a bijection between a proper subset of A and A itself thus leading to a contradiction. My only concern is that I'm not sure how to show with rigor these steps. Most notably how to construct a proper subset A such that f is one to one and onto A.

 

[morphism]

You don't need the axiom of choice. The fundamental property here is the finiteness of A.

 

[moo5003]

Suppose f is onto.

Suppose by contradiction f is not injective.

Let g be a proper subset of f such that g is one-to-one and onto A. Since f was not injective this implies g is a proper subset of f and the domain of g = B is a proper subset of A.

Thus:
f[A]=A onto not injective
g=A bijection

This implies card(B) = card(A) which is a contradiction since A is finite and B is a proper subset of A implying card(B) < card(A)

Is there anything wrong with this?