# Types of Maps * A [[Map|map]] $f: X \rightarrow Y$ is *injective* or (*one-to-one*, or an *injection*) if it maps an element in its [[Domain, Co-domain, Image|domain]] to a unique element in its [[Domain, Co-domain, Image|range]], i.e., if $x \neq x'$ then $f(x) \neq f(x')$. * A [[map]] $f: X \rightarrow Y$ is from $X$ *into* $Y$ if it is defined for all points in its domain. * A [[map]] $f:X\rightarrow Y$ is called *surjective* or (*onto*, or a *surjection*) if it is *into*, and the image of its domain is [[Isomorphism|isomorphic]] to its [[Domain, Co-domain, Image|co-domain]] ($\text{im}(X) \cong Y$). In other words, a surjection maps all points in its domain to all points in its co-domain, in a many-to-one way, [$\forall y \in Y \exists x \in X : f(x) = y$]) * A [[map]] $f:X \rightarrow Y$ is called a *bijection* if it is both an *injection* and a *surjection*. If a map is a bijection, then an [[Inverse Map]] exists. See the figure below for a simple diagram. ![[Pasted image 20210122200039.png]]