Added section to maps in set theory.

docs/en/mathematics/set-theory/maps.md

@@ -43,6 +43,29 @@ If $B'$ is a subet of $B$ then the pre-image of $B'$, denoted by $f^{-1}(B') is
 
 ??? note "*Proof*:"
 
-   Let $a' \in A'$, then $f(a') \in f(A')$ and hence $a' \in f^{-1}(f(A'))$. Thus $A' \subseteq f^{-1}(f(A'))$.
+    Let $a' \in A'$, then $f(a') \in f(A')$ and hence $a' \in f^{-1}(f(A'))$. Thus $A' \subseteq f^{-1}(f(A'))$.
+
+    Let $a \in f^{-1}(B')$, then $f(a) \in B'$. Thus $f(f^{-1}(B')) \subseteq B'$.
+
+## Special maps
+
+> *Definition*: let $f: A \to B$ be a map.
+>
+> * $f$ is called **surjective**, if for each $b \in B$ there is at least one $a \in A$ with $b = f(a)$. Thus $\text{Im}(f) = B$.
+> * $f$ is called **injective** if for each $b \in B$, there is at most one $a$ with $f(a) = b$.
+> * $f$ is called **bijective** if it is both surjective and injective. So, if for each $b \in B$ there is a unique $a \in A$ with $f(a) = b$.
+
+For example the map $\sin: \mathbb{R} \to \mathbb{R}$ is not surjective nor injective. The map $\sin: [-\frac{\pi}{2},\frac{\pi}{2}] \to \mathbb{R}$ is injective but not surjective and the map $\sin: \mathbb{R} \to [-1,1]$ is surjective but not injective. To conclude the map $\sin: [-\frac{\pi}{2},\frac{\pi}{2}] \to [-1,1]$ is a bijective map.
+
+<br>
+
+> *Theorem*: let $A$ be a set of size $n$ and $B$ a set of size $m$. Let $f: A \to B$ be a map between the sets $A$ and $B$.
+>
+> * If $n < m$ then $f$ can not be surjective.
+> * If $n > m$ then $f$ can not be injective.
+> * If $n = m$ then $f$ is injective if and only if it is surjective.
+
+??? note "*Proof*:"
+
+    Think of pigeonholes. (Not really a proof).
 
-   Let $a \in f^{-1}(B')$, then $f(a) \in B'$. Thus $f(f^{-1}(B')) \subseteq B'$.