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?
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
hint: $\displaystyle \aleph _0 \cdot \aleph _0$ is the cardinal number of $\displaystyle \mathbb{N} \times \mathbb{N}$ by definition. you can prove that this cardinal number is also infinitely countable, which will show that it is equal to $\displaystyle \aleph _0$. 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 $\displaystyle \mathbb{N}$ along the first row and column, and the elements in the inner cells are there corresponding products. you count off the elements in diagonals ...or
Xo * Xo = Xo
Help anyone?