One to One Function and Onto Function
The key differences between One to One and Onto Functions are listed in the following table:
| Property | One-to-One (Injective) Function | Onto (Surjective) Function |
|---|---|---|
| Definition | A function in which no two different elements in the domain map to the same element in the codomain. In other words, each element in the domain maps to a unique element in the codomain. | A function in which every element in the codomain is mapped to by at least one element in the domain. In other words, the range of the function equals the entire codomain. |
| Symbolic Representation | f(x1) ≠ f(x2) if x1 ≠ x2 for all x1, x2 in the domain. | For every y in the codomain, there exists an x in the domain such that f(x) = y. |
| Graphical Representation | The graph of a one-to-one function never has a horizontal line that intersects it at more than one point. | The graph of an onto function may not cover every point on the codomain, but it covers every point that it can, meaning there are no “gaps” in the codomain. |
| Example | f(x) = 2x is one-to-one because no two distinct values of x produce the same output. | f(x) = √x is onto for non-negative real number as its codomain because, all non-negative real numbers has a preimage in this function. |
| Inverse Function | A one-to-one function generally has an inverse function. | An onto function may or may not have an inverse function. |
| Cardinality | The cardinality of the domain and codomain can be equal or different for one-to-one functions. | The cardinality of the codomain is usually greater than or equal to the cardinality of the domain for onto functions. |
The following illustration provide the clear difference between one one and onto function:
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One to One Functions in Mathematics
One to One Function or One-One Function is one of the types of functions defined over domain and codomain and describes the specific type of relationship between domain and the codomain. One to One Function is also called the Injective Function. One to One Function is a mathematical function where each element in the domain maps to a unique element in the codomain.
This article explores the concept of One to One Function or One-One Function in detail including its definition and examples which help you understand the concept with ease. We will also discuss some sample problems and provide some practice problems for you to solve. So, let’s learn about this important concept in mathematics known as One to One Function.
Table of Content
- What is One-to-One Function?
- Examples of One-to-One Functions
- Properties of One-to-One Functions
- One to One Function and Onto Function
- Solved Examples on One to One Function