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Identity and Invertible Functions in Discrete Mathematics

Explore identity and invertible functions in discrete mathematics. This guide defines identity functions, explains invertible functions (bijections), and shows how to determine if a function is invertible. Examples are provided to illustrate these concepts.

Identity and Invertible Functions in Discrete Mathematics

Identity Functions

An identity function is a simple type of function where each input value maps to itself. In other words, the function leaves the input unchanged. For a function f on a set A, it's an identity function if f(a) = a for every element a in A. Identity functions are usually denoted by the symbol I.

Example: Identity Function

Let A = {1, 2, 3, 4, 5}. The function f: A → A defined as:

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

is an identity function. This function is both one-to-one (injective) and onto (surjective).

Invertible Functions

A function is invertible if it has an inverse function. A function is invertible if and only if it's bijective (both one-to-one and onto). This means:

  • One-to-one (injective): Every element in the domain maps to a unique element in the codomain (no two inputs produce the same output).
  • Onto (surjective): Every element in the codomain is mapped to by at least one element in the domain (every possible output is used).

Example: Invertible Function

Let X = {1, 2, 3} and Y = {k, l, m}. The function f: X → Y defined as:

f = {(1, k), (2, m), (3, l)}

is invertible because it's both one-to-one and onto. The inverse function, f-1, maps each element in Y back to its unique corresponding element in X.

Inverse Functions

The inverse of function f (denoted f-1) reverses the mapping of the original function. It only exists if f is bijective. If f(a) = b, then f-1(b) = a.

(The example from the original text showing a function and its inverse is given here.)

Conclusion

Identity functions are the simplest functions where each input maps to itself. Invertible functions are a special class of functions that are both one-to-one and onto, and because of these properties, they always have an inverse.