Bijection, injection, surjection

Các khái niệm toán học thường dùng trên tập hợp:
1. Bijection: song ánh.
2. Injection: đơn ánh.
3. Surjection: toàn ánh.

Bijection

A transformation which is one-to-one and a surjection (i.e., "onto").

Injection

Let  be a function defined on a set  and taking values in a set . Then  is said to be an injection (or injective map, or embedding) if, whenever , it must be the case that . Equivalently,  implies . In other words,  is an injection if it maps distinct objects to distinct objects. An injection is sometimes also called one-to-one.

A linear transformation is injective if the kernel of the function is zero, i.e., a function  is injective iff .

A function which is both an injection and a surjection is said to be a bijection.

In the categories of sets, groups, modules, etc., a monomorphism is the same as an injection, and is used synonymously with "injection" outside of category theory.

Surjection

Let  be a function defined on a set  and taking values in a set . Then  is said to be a surjection (or surjective map) if, for any , there exists an  for which . A surjection is sometimes referred to as being "onto."

Let the function be an operator which maps points in the domain to every point in the range and let  be a vector space with . Then atransformation  defined on  is a surjection if there is an  such that  for all .

In the categories of sets, groups, modules, etc., an epimorphism is the same as a surjection, and is used synonymously with "surjection" outside ofcategory theory.