Some terminology of set theory

$${\mathbb{N}} := \{0,1,2,3,\dots\}$$

$${\mathbb{Z}} := \{\dots,-2,-1,0,1,2,\dots\}$$

$${\mathbb{Q}} :=$$

$${\mathbb{R}} :=$$

$${\mathbb{C}} :=$$

(Some authors prefer not to include 0 in ${\mathbb{N}}$, but including 0 is more natural for many purposes; for example, then ${\mathbb{N}}$ is exactly the set of possibilities for the size of a finite set. Other authors avoid the issue entirely by using ${\mathbb{Z}}_{>0}$ to denote the set of positive integers and ${\mathbb{Z}}_{\ge 0}$ to denote the set of nonnegative integers.)

Let $f:X \rightarrow Y$ be a function from a set $X$ to a set $Y$. Then

$$f \iff f(x_1) \not= f(x_2) x_1 \not= x_2$$

$$f \iff y \in Y x \in X f(x)=y$$

$$f \iff f$$

In the last case, one also says that $f$ is a bijection (one-to-one correspondence); then $f$ pairs the elements of $X$ with the elements of $Y$ in such a way that no elements of either set are left unpaired.