# Math Help - axiom of choice

1. ## axiom of choice

Define an injective map $f: \mathbb{Z}^+ \rightarrow \{0,1\}^\omega$ without using the axiom of choice.

I would define an injective function by $f(n)=x$, where $x_i = 1$ if $i=n$ and $x_i=0$ if $i \not= n$. But how would I define the map without using the axiom of choice?

2. Originally Posted by dori1123
Define an injective map $f: \mathbb{Z}^+ \rightarrow \{0,1\}^\omega$ without using the axiom of choice.

I would define an injective function by $f(n)=x$, where $x_i = 1$ if $i=n$ and $x_i=0$ if $i \not= n$. But how would I define the map without using the axiom of choice?
If $n>0$ let $n\mapsto \left< 1,\underbrace{1,1,...,1}_{n\text{ times }},0,0,0,... \right>$

If $n<0$ let $n\mapsto \left< 0, \underbrace{1,1,...,1}_{|n|\text{ times }},0,0,0,... \right>$

If $n=0$ let $n\mapsto \left<0,0,0,0,0,0,...\right>$