User:Sdoroudi/Sandbox

Math scratchwork.

Theorem 1 $$\operatorname{card}(\mathbb N)=\operatorname{card}\left(\mathbb N^2\right)$$
Let

$$\varphi\colon\mathbb N^2\to\mathbb N$$

be the map defined by

$$\varphi(a,b)\mapsto 2^a(2b-1)$$.

It follows from the fundemental theorem of arithmetic that $$\varphi$$ is a bijection.

Hence, we have

$$\operatorname{card}(\mathbb N)=\operatorname{card}\left(\mathbb N^2\right)$$.

Theorem 2 $$\operatorname{card}(\mathbb N)=\operatorname{card}(\mathbb Z)$$
Let

$$\phi\colon\mathbb Z\to\mathbb N$$

be the map defined by

$$\phi(k)\mapsto 2|k|+\mu(k)$$,

where $$\mu$$ is the Heaviside function.

It is easily shown that $$\phi$$ is a bijection.