Functions in Discrete Mathematics: Domain, Codomain, and Range

Learn about functions in discrete mathematics, focusing on their definition as mappings between sets. This guide explains the concepts of domain, codomain, and range, and provides examples to illustrate the properties and characteristics of functions.

Understanding Functions in Discrete Mathematics

What is a Function?

A function is a special type of relation between two sets, a domain (inputs) and a codomain (possible outputs), where each input is associated with exactly one output. Think of a function like a machine: you put something in (the input), and it gives you something out (the output). The output is uniquely determined by the input.

Key Function Terminology

  • Domain: The set of all possible input values.
  • Codomain: The set of all possible output values.
  • Range: The set of actual output values produced by the function (a subset of the codomain).

Mathematical Representation of a Function

A function f from set P to set Q (written f: P → Q) maps each element x in P to a unique element y in Q. The range of f is the set of all its outputs and is written:

f(P) = {f(x) | x ∈ P} = {y | y ∈ Q and ∃x ∈ P such that f(x) = y}

Example: Identifying Domain, Codomain, and Range

(The example from the original text defining sets X and Y and function f, along with the identification of its domain, codomain, and range, should be included here.)

Functions as Sets of Ordered Pairs

A function f from set P to set Q can be represented as a set of ordered pairs (p, q) where p ∈ P, q ∈ Q, and p is mapped to q under f. Two key rules apply:

  1. Every element in P must appear as the first element of exactly one ordered pair.
  2. If (a, b) ∈ f and (a, c) ∈ f, then b = c (each input maps to only one output).

Example: Number of Functions

If set A has n elements, then there are nn functions from A to itself.

Representing Functions Graphically

(An illustrative example using sets X and Y and function f showing the graphical representation of the function using arrows is given in the original text and should be included here.)

Example: Determining if Relations are Functions

(This example from the original text testing several relations (f, g, h, l, d) to determine if they are functions, finding the range for functions, and explaining why the others are not functions should be included here.)

Conclusion

Functions are fundamental mathematical objects. This overview defines essential terminology and illustrates how to determine if a relation is a function.