What is the other name of injective?

Injective functions, also known as injections or one-to-one functions, play a fundamental role in mathematics. These functions are characterized by their unique mapping property.

Specifically, an injective function, denoted as f, maps each distinct element of its domain to a distinct element of its range. This means that if two elements in the domain are not equal, their images under the function will also be different.

Formally, for any two elements x1 and x2 in the domain of f, if x1 is not equal to x2, then f(x1) must not be equal to f(x2). This property ensures that every element in the range of f corresponds to exactly one element in its domain.

The concept of injectivity is essential in understanding the behavior of functions and their inverses. An injective function guarantees the existence of a left inverse, which is a function that "undoes" the effect of f.

Injective functions are often encountered in various mathematical contexts, such as in set theory, algebra, and analysis. They play a crucial role in proving theorems and constructing proofs.