1. ## cardinality

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

thats it thanks

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

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

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

May be you mean:

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

CB

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

4. oops I meant X+N = X

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

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

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