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?

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- Jan 22nd 2008, 09:23 AMtasheCardinal 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, 09:49 AMJhevon
i assume $\displaystyle k$ is a cardinal number such that $\displaystyle k \le \aleph _0$. otherwise, it is not true

we can prove the claim by showing:

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

(1) is immediate.

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

Quote:

or

Xo * Xo = Xo

Help anyone?