4.12.1.1–4.12.1.3 - Function types, first-class values and application (VB.NET)

Domain, co-domain and function types

A function assigns exactly one output to each permitted input. Its type records the sets from which those inputs and outputs are drawn.

For example, let $A=\{-2,-1,0,1,2\}$, $B=\{0,1,2,3,4\}$, and $f(x)=x^2$. This defines $f:A\rightarrow B$. All five permitted inputs have one output, but the outputs actually reached are only $\{0,1,4\}$. This reached set is the range. The unused values 2 and 3 can still belong to the co-domain. Different inputs may have the same output: both −2 and 2 map to 4.

A second example maps a lowercase letter to its alphabetical position: $p:\{a,b,\ldots,z\}\rightarrow\{0,1,\ldots,25\}$, with $p(a)=0$ and $p(z)=25$. Here every co-domain value is reached. That is possible, but it is not required of every function.

The declaration Func(Of Integer, Integer) describes a VB.NET function value taking an Integer and returning an Integer. It describes data types, not a narrower permitted set such as $A$. A task must state any input restriction separately. Mathematical integers are unbounded; VB.NET Integer values are bounded. Our small examples stay within their range.

First-class function values

A first-class value can appear in expressions, be assigned to a variable, be passed as an argument, and be returned from a function. Integers and strings commonly have these capabilities; functions have them too in functional languages and in languages such as VB.NET that support function values.

A VB.NET delegate is a value referring to callable code. Func(Of Integer, Integer) is a delegate type whose final type parameter is the result type. A lambda expression creates an unnamed function: Function(n As Integer) n * 2 returns twice its argument. A single-line lambda contains its result expression without Return or End Function.

Imports System

Module Program
    Function ChooseTransform() As Func(Of Integer, Integer)
        Return Function(n As Integer) n * 2
    End Function

    Function ApplyTwice(transform As Func(Of Integer, Integer),
                        value As Integer) As Integer
        Return transform(transform(value))
    End Function

    Sub Main()
        Dim doubleValue As Func(Of Integer, Integer) = ChooseTransform()
        Dim anotherName As Func(Of Integer, Integer) = doubleValue
        Console.WriteLine(ApplyTwice(anotherName, 3))
    End Sub
End Module