## Links forward

### Functions between finite sets

If \(A\) and \(B\) are finite sets, then a function from \(A\) to \(B\) can be specified by listing all elements in \(A\) and their images in \(B\). For example, if

\[ A = \{a_1,a_2,a_3,a_4\} \qquad\text{and}\qquad B = \{b_1,b_2,b_3,b_4,b_5\}, \] then we can define a function \(f\colon A\to B\) by \[ f(a_1)=b_4, \quad f(a_2)=b_1, \quad f(a_3)=b_5 \quad\text{and}\quad f(a_4)=b_5. \]This can be represented by an **arrow diagram**.

For this example, we have \(\mathrm{domain}(f)=A\), \(\mathrm{codomain}(f)=B\) and \(\mathrm{range}(f)=\{b_1,b_4,b_5\}\).