Mapping/Graph/Image/Preimage/Introduction/Section

Definition

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

$F\colon L\longrightarrow M$

be a mapping. Then the set

${}\Gamma =\Gamma _{F}={\left\{(x,F(x))\mid x\in L\right\}}\subseteq L\times M\,$

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

The graph is a concept of set theory. Whether it is possible to "visualize“ it in a picture depends on whether we can visualize the product set ${}L\times M$.

Definition

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

$F\colon L\longrightarrow M$

be a mapping. For a subset ${}S\subseteq L$, we call

${}F(S)={\left\{y\in M\mid {\text{there exists an }}x\in S{\text{ such that }}F(x)=y\right\}}\,$

the image of ${}S$ under ${}F$. For ${}S=L$,

${}F(L)=\operatorname {Im} F\,$

is called the image of the mapping.

Definition

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

$F\colon L\longrightarrow M$

be a mapping. For a subset ${}T\subseteq M$, we call

${}F^{-1}(T)={\left\{x\in L\mid F(x)\in T\right\}}\,$

the preimage of ${}T$ under ${}F$. For a subset ${}T=\{y\}$ with one element, we call

$F^{-1}(\{y\})$

also the preimage of ${}y$.