Function

Given two set $W,U$, $W\times U=\{(w,u)|w\in W,u\in U\}$

Definition

A function $f:X\to Y$ is a subset $f\subset X\times Y$ such that for each element $x\in X$, there exists a unique element $y\in Y$ where $(x,y)\in f$.

• $X$ is called as domain of $f$.
• $Y$ is called as codomain of $f$.

Range

The range of $f:X\to Y$ is the set of $y\in Y$ such that $y=f(x)$ for some $x\in X$

Injectivity

$f:X\to Y$ is injective (also called one-by-one) if $f(x_1)=f(x_2)$ implies $x_1=x_2.$ For injective function, we can define the inverse function, denoting as $f^{-1}$.