Functions in Discrete Mathematics: Definition, Domain, Codomain, and Range

This guide provides a comprehensive explanation of functions in discrete mathematics, covering key concepts such as domain, codomain, range, and the conditions that define a function. Numerous examples are used to illustrate these concepts.

Understanding Functions in Discrete Mathematics

Introduction to Functions

In mathematics, a function is a special type of relation between two sets—a domain (the set of inputs) and a codomain (the set of possible outputs). For each input value in the domain, a function assigns exactly one output value in the codomain. This means that if you give a function the same input twice, you will always get the same output.

Domain, Codomain, and Range

Let's define key terms:

  • Domain: The set of all possible input values for a function.
  • Codomain: The set of all possible output values for a function.
  • Range: The set of *actual* output values produced by the function for the given inputs. The range is always a subset of the co-domain.

Example: Domain, Codomain, and Range

Let X = {1, 2, 3, 4} and Y = {a, b, c, d, e}. Consider the function f = {(1, b), (2, a), (3, d), (4, c)}.

  • Domain: {1, 2, 3, 4}
  • Range: {a, b, c, d}
  • Codomain: {a, b, c, d, e}

Functions as Sets

A function f from set P to set Q (written as f: P → Q) can be represented as a subset of the Cartesian product P x Q. Two conditions must be met:

  1. For every element a in P, there's an element b in Q such that (a, b) is in f. This means that every input in the domain has a corresponding output.
  2. If (a, b) is in f and (a, c) is in f, then b = c. This is the uniqueness condition; each input can only have one output.

Example: Number of Functions from a Set to Itself

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

Representing Functions Graphically

Functions can be visualized using a graph. The domain and codomain are represented as sets, and arrows connect elements in the domain to their corresponding elements in the codomain.

(An example diagram showing a function represented graphically would be very helpful here.)

Example: Determining if a Relation is a Function

Let X = {x, y, z, k} and Y = {1, 2, 3, 4}. Determine if these relations are functions, and identify the range if they are:

  1. f = {(x, 1), (y, 2), (z, 3), (k, 4)}
  2. g = {(x, 1), (y, 1), (k, 4)}
  3. h = {(x, 1), (x, 2), (x, 3), (x, 4)}
  4. l = {(x, 1), (y, 1), (z, 1), (k, 1)}
  5. d = {(x, 1), (y, 2), (y, 3), (z, 4), (z, 4)}

(Solutions and explanations for each relation should be included here.)

Conclusion

Functions are a fundamental concept in mathematics. Understanding their properties, how they're represented, and how to determine if a relation is a function is essential for various mathematical and computational tasks.