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.