# Math Help - cardinality

1. ## cardinality

if $X$ is any infinite set show that $|X| + |\mathbb{N}| = |\mathbb{N}|$

thats it thanks

2. Originally Posted by jbpellerin
if $X$ is any infinite set show that $|X| + |\mathbb{N}| = |\mathbb{N}|$

thats it thanks
Not true, with the wording you have $X=\mathbb{R}$ is permitted, and

$|\mathbb{R}| + |\mathbb{N}| = |\mathbb{R}| \ne |\mathbb{N}|$

May be you mean:

if $X$ is any finite set show that $|X| + |\mathbb{N}| = |\mathbb{N}|$

CB

3. Or if $X$ is an infinite set, $|X|+|\mathbb{N} |=|X|$

4. oops I meant X+N = X

5. Well, that depends on what you know about infinity.

If you know there exists a $B \subset X$ such that $|B|=|\mathbb{N}|$, you can find a bijection between $B\cup \mathbb{N}$ and $\mathbb{N}$ (that just means $B\cup \mathbb{N}$ is countable), and you've won.

Indeed, you get $X \equiv (X-B)\cup B \equiv (X-B)\cup \mathbb{N} \equiv (X-B)\cup (B\cup \mathbb{N}) \equiv X\cup \mathbb{N}$
(here $\equiv$ means "equipotent to")