Mapping/Injective/Surjective/Bijective/Introduction/Section

Definition

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called injective if for two different elements ${}x,x'\in L$, also ${}F(x)$ and ${}F(x')$

are different.

Definition

Let ${}L$ and ${}M$ denote sets, and let

$F\colon L\longrightarrow M,x\longmapsto F(x),$

be a mapping. Then ${}F$ is called surjective if, for every ${}y\in M$, there exists at least one element ${}x\in L$, such that

${}F(x)=y\,.$

Definition

Let ${}M$ and ${}L$ denote sets, and suppose that

$F\colon M\longrightarrow L,x\longmapsto F(x),$

is a mapping. Then ${}F$ is called bijective if ${}F$ is injective as well as

surjective.

These concepts are fundamental!

The question, whether a mapping ${}F\colon L\rightarrow M$ has the properties of being injective or surjective, can be understood looking at the equation

${}F(x)=y\,$

(in the two variables ${}x$ and ${}y$). The surjectivity means that for every ${}y\in M$ there exists at least one solution

${}x\in L\,$

for this equation; the injectivity means that for every ${}y\in M$ there exists at most one solution ${}x\in L$ for this equation. The bijectivity means that for every ${}y\in M$ there exists exactly one solution ${}x\in L$ for this equation. Hence, surjectivity means the existence of solutions, and injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics, and they can also be interpreted as surjectivity or injectivity of suitable mappings.

In order to show that a certain mapping is injective, we often use the following strategy: One shows for any two given elements ${}x$ and ${}x'$ using the condition ${}F(x)=F(x')$ that ${}x=x'$ holds. This method is often easier than showing that ${}x\neq x'$ implies ${}F(x)\neq F(x')$.

Example

The mapping

$\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is neither injective nor surjective. It is not injective because the different numbers ${}2$ and ${}-2$ are both sent to ${}4$. It is not surjective because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} ,x\longmapsto x^{2},$

is injective, but not surjective. The injectivity can be seen as follows: If ${}x\neq y$, then one number is larger, say

${}x>y\geq 0\,.$

But then also ${}x^{2}>y^{2}$, and in particular ${}x^{2}\neq y^{2}$. The mapping

$\mathbb {R} \longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is not injective, but surjective, since every nonnegative real number has a square root. The mapping

$\mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},x\longmapsto x^{2},$

is injective and surjective.

Definition

Let ${}F\colon L\rightarrow M$ denote a bijective mapping. Then the mapping

$G\colon M\longrightarrow L$

that sends every element ${}y\in M$ to the uniquely determined element ${}x\in L$ with ${}F(x)=y$,

is called the inverse mapping of ${}F$.

The inverse mapping is usually denoted by ${}F^{-1}$.