The cardinality of ${\mathbb{Z}}$

Suppose we want to check whether the set ${\mathbb{Z}}$ of integers has the same cardinality as ${\mathbb{N}}$. If we try to set up a bijection from ${\mathbb{N}}$ to ${\mathbb{Z}}$ without thinking, we fail because the negative numbers are not used:

$$\longleftrightarrow 0$$ 0

$$1 \longleftrightarrow 1$$

$$2 \longleftrightarrow 2$$

$$3 \longleftrightarrow 3$$

$$\vdots$$

$$\qquad -1,-2,-3,\dots$$

This is only an injection. Does this mean that $\char93 {\mathbb{N}}\not= \char93 {\mathbb{Z}}$? No! Even though this function did not give a bijection, it is easy to construct other functions ${\mathbb{N}}\rightarrow {\mathbb{Z}}$ that are bijections, like

$$\longleftrightarrow 0$$ 0

$$1 \longleftrightarrow 1$$

$$2 \longleftrightarrow -1$$

$$3 \longleftrightarrow 2$$

$$4 \longleftrightarrow -2$$

$$5 \longleftrightarrow 3$$

$$6 \longleftrightarrow -3$$

$$\vdots$$

Thus $\char93 {\mathbb{Z}}= \char93 {\mathbb{N}}= \aleph_0$, even though ${\mathbb{N}}$ is a proper subset of ${\mathbb{Z}}$. (This is similar to the situation in the previous section.)