# Cardinal numbers proof

• Jan 22nd 2008, 10:23 AM
tashe
Cardinal numbers proof
How can I prove these lines:

Let N be a set of natural numbers and |N|=Xo.
How can i prove that
Xo + k = Xo
or
Xo * Xo = Xo

Help anyone?
• Jan 22nd 2008, 10:49 AM
Jhevon
Quote:

Originally Posted by tashe
How can I prove these lines:

Let N be a set of natural numbers and |N|=Xo.
How can i prove that
Xo + k = Xo

i assume $k$ is a cardinal number such that $k \le \aleph _0$. otherwise, it is not true

we can prove the claim by showing:

(1) $\aleph _0 + k \ge \aleph _0$ and (2) $\aleph _0 + k \le \aleph _0$

(1) is immediate.

for (2): $\aleph _0 + k \le \aleph _0 + \aleph _0 = 2 \aleph _0 = \aleph _0$ ....there should be a theorem in your book telling you something to that effect

Quote:

or
Xo * Xo = Xo

Help anyone?
hint: $\aleph _0 \cdot \aleph _0$ is the cardinal number of $\mathbb{N} \times \mathbb{N}$ by definition. you can prove that this cardinal number is also infinitely countable, which will show that it is equal to $\aleph _0$. the way i usually see this done is by the use of a diagram, you should also see a proof using this technique in your text. the diagram looks like a grid, with the elements of $\mathbb{N}$ along the first row and column, and the elements in the inner cells are there corresponding products. you count off the elements in diagonals ...