Functions
Table of Contents
1. Function Definition
A function is a mapping from a domain \( A \) to a codomain \( B \). More specifically, a function associates every value in the domain with precisely one value in the codomain.
\[ f\colon A\to B : x\mapsto f(x) \]
A function is bijective if it is both injective and surjective.
1.1. Examples
| Injective | Not Injective | |
|---|---|---|
| Surjective | \( f\colon \mathbb{R} \to \mathbb{R} : x\mapsto x \) | \( f\colon \mathbb{R} \to \mathbb{R} : x\mapsto x^3 + x \) |
| Not Surjective | \( f\colon \mathbb{R} \to \mathbb{R} : x\mapsto e^x \) | \( f\colon \mathbb{R} \to \mathbb{R} : x\mapsto x^2 \) |
2. Injective Property
A function is injective if it describes a one-one mapping between \( A \) and \( B \). In other words every element in the image f is associated with precisely one value in the domain \( A \).
If \( f \) is injective then \( |A| = |f(A)| \)
3. Surjective Property
A function is surjective if the image of the function is equal to its codomain. If this is true then \( |f(A)| = |B| \).
4. Permutation
Let \( A \) be a set, then a permutation is a bijection from \( A\to A \). Hence \( f\colon A \to A \) is a permutation if it is injective (and therefore surjective).