# Math Help - Show that every denumerable set is equivalent to a proper subset of itself

1. ## Show that every denumerable set is equivalent to a proper subset of itself

I'm not sure how to do this. Help would be appreciated, thanks.

2. If $A$ is denumerable, there is a bijection $f$ between $A$ and $\mathbb{N}$. Take any $a\in A$. Show that $\displaystyle\mathbb{N}$ and $\mathbb{N}\setminus\{f(a)\}$ are equinumerous. Without loss of generality, one may assume that $f(a)=0$.

3. Originally Posted by dynas7y
I'm not sure how to do this. Help would be appreciated, thanks.

If a set A is denumerable (meaning: INFINITE denumerable), then we can write $A=\{a_1,a_2,a_3,\ldots\}$, so define

$f:A\rightarrow A-\{a_1\}\,,\,\,f(a_i):=a_{i+1}$ . Now just prove $f$ is a bijection...

Tonio