What is the difference between one-to-one and injective?

A function is considered injective, or one-to-one, when each element in the codomain of the function is associated with at most one element from the domain. This property ensures that the function does not map two or more distinct elements from the domain to the same element in the codomain.

Injectivity is a crucial characteristic of functions, as it allows for a unique relationship between the domain and the codomain. When a function is injective, every input has a unique output, and no two inputs can result in the same output.

An alternative way to understand injectivity is to consider how the function maps distinct arguments to distinct images. In other words, if two inputs are different, their outputs must also be different for the function to be injective.

The term "injection" is often used interchangeably with "injective function" to describe this unique mapping property. An injection ensures that there is no loss of information during the function's operation, as each input is mapped to a distinct output.

The concept of injectivity is essential in various mathematical and computational contexts, including cryptography, coding theory, and optimization problems. It is also a fundamental concept in set theory, where injections are used to define the cardinality of sets.