[[Naïve set theory MOC]]
# Surjectivity, injectivity, and bijectivity

**Surjective**, **injective**, and **bijective** functions are [[Morphism|epimorphisms]],  [[Morphism|monomorphisms]], and [[Morphism|isomorphisms]] respectively in [[Category of sets]].
Thus the morphisms of any [[concrete category]] may be described as such,
but these concepts may not align exactly
(for example, there exist bijectivity continuous functions that are not homeomorphisms).
Specifically, given a function $f:A \to B$

- $f$ is **surjective** iff for every $b \in B$ there exists $a \in A$ such that $f(a) = b$ #m/def/general 
    - Equivalently, there exists a right-inverse.
    - A surjective function induces an [[Equivalence relation]].
- $f$ is **injective** iff $f(a_{1}) = f(a_{2}) \iff a_{1} = a_{2}$. #m/def/general
  - Equivalently, there exists a left-inverse.
- $f$ is **bijective** iff it is surjective and injective. #m/def/general 
  - Equivalently, there exists a unique ambidextrous inverse.

#
---
#state/tidy | #lang/en | #SemBr