---- > [!definition] Definition. ([[graph]]) > Let $f:A \to B$ be a function. The **graph** of $f$ is the subset $\Gamma(f):=\{ (a,b) \in A \times B : b=f(a) \} \subset A \times B.$ ^definition > [!note] Remark. > Officially, a function really 'is' its graph as far as set theory goes. But one usually prefers to think of functions as 'verbs' and graphs as 'nouns'. ^note > [!basicexample] > For a function $f: \mathbb{R}^n \to\mathbb{R}$, the graph of $f$ is the subset of $\mathbb{R}^{n+1}$ given by $\text{Gr}_f = \{ (x_1, \dots, x_{x_n+1}) : x_{n+1} = f(x_1, \dots, x_n) \}.$ \ This is the set we conventionally visualize when we say we are 'graphing' a function. ^basic-example > [!basicproperties] > - $\text{Gr}_{f}$ is isomorphic to (i.e., in [[bijection|bijective correspondence with]])the domain $A$ itself. The [[bijection]] is $a \mapsto (a,f(a))$; this is an [[injection]] because if $(a,f(a))=(b,f(b))$ then in particular $f(a)=f(b)$ and the function definition consequently enforces $a=b$, and it is a [[surjection]] because an arbitrary $(a, f(a)) \in \text{Gr}_{f}$ gets mapped to by $a \in A$. ^properties ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```