Results 1 to 2 of 2

Thread: axiom of choice

  1. #1
    Member
    Joined
    Sep 2008
    Posts
    166

    axiom of choice

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

    I would define an injective function by $\displaystyle f(n)=x$, where $\displaystyle x_i = 1$ if $\displaystyle i=n$ and $\displaystyle x_i=0$ if $\displaystyle i \not= n$. But how would I define the map without using the axiom of choice?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by dori1123 View Post
    Define an injective map $\displaystyle f: \mathbb{Z}^+ \rightarrow \{0,1\}^\omega$ without using the axiom of choice.

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

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

    If $\displaystyle n=0$ let $\displaystyle n\mapsto \left<0,0,0,0,0,0,...\right>$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Axiom of choice.
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: Jan 11th 2010, 08:50 AM
  2. With or without Choice-axiom?
    Posted in the Discrete Math Forum
    Replies: 11
    Last Post: Dec 19th 2009, 07:08 PM
  3. Axiom of choice
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Feb 18th 2009, 02:27 AM
  4. Axiom of Choice
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Jan 18th 2006, 01:52 PM

Search Tags


/mathhelpforum @mathhelpforum