Identity Functions in Discrete Mathematics: Definition and Properties

Learn about identity functions, functions where the output is always equal to the input. This guide provides a formal definition, explains their properties, and illustrates identity functions with examples.

Identity Functions in Discrete Mathematics

What is an Identity Function?

An identity function is a simple yet important type of function. It's a function where the output is always the same as the input. In other words, the function leaves the input unchanged.

Definition of an Identity Function

Formally, a function g is an identity function if g(x) = x for all x in its domain. The identity function is often denoted by the symbol I. The domain and range of an identity function are the same.

Example of an Identity Function

Let's say we have a set A = {1, 2, 3, 4, 5}. An identity function g: A → A would be defined as:

g = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}

Notice how each element maps to itself.

Domain, Range, and Inverse of an Identity Function

For an identity function g: ℝ → ℝ (where ℝ is the set of real numbers):

  • The domain is the set of real numbers (ℝ).
  • The range is also the set of real numbers (ℝ).
  • The function is both one-to-one (injective) and onto (surjective).
  • The inverse of the identity function is itself (g-1(x) = x).

Graph of the Identity Function

The graph of an identity function, y = x, is a straight line passing through the origin (0,0) with a slope of 1. It forms a 45-degree angle with both the x and y axes.

(An illustrative graph of y = x would be included here.)

Properties of the Identity Function

  • It's a linear operator on vector spaces.
  • It's a multiplicative function on positive integers.
  • It's a real-valued linear function.
  • Its graph is identical to the graph of its inverse.
  • In an m-dimensional vector space, it's represented by the identity matrix Im.
  • It's always continuous in topological spaces.

Important Notes on Identity Functions

  • The domain and range are always the same.
  • The slope of its graph is always 1.

Examples of Identity Functions

Example 1: Composition of a Function with Itself

Given g(y) = (2y + 3) / (3y - 2), show that g(g(y)) = y (meaning g composed with itself is the identity function).

(The calculation demonstrating g(g(y)) = y would be included here.)

Example 2: A Simpler Identity Function

Show that f(2x) = 2x is an identity function.

(A table showing sample inputs and outputs demonstrating that f(2x) = 2x is an identity function would be included here. A graph of this function would also be appropriate.)

Example 3: Identity Function on a Set

If we have a set with 9 elements, what is the appropriate size of the range of the identity function on this set?

The answer is 9 (since the range is equal to the domain for an identity function).

Conclusion

The identity function, while seemingly simple, is a fundamental concept with significant implications in various branches of mathematics.