ece108: add injective/surjective

docs/1b/ece108.md

@@ -516,3 +516,31 @@ $$\text{preimage}(f)=\{x\in X|\exists y\in B,y=f(x)\}$$
 The **image** is the subset of the codomain that is mapped by a specific subset $A$ of the domain.
 
 $$\text{image}(f)=\{f(x)|\exists x\in A\}$$
+
+!!! example
+    For the function $f: \mathbb R^+_0\to \mathbb R$ defined by $x\longmapsto x^2$:
+    
+    - the domain is $\mathbb R^+_0$
+    - the codomain is $\mathbb R$
+    - the range is $\mathbb R^+_0$
+    - the preimage for $\{1\}$ is $\{1,-1\}$
+    - the image for $0$ is $\{0\}$
+
+Two functions $f=g$ are equal if and only if:
+
+- their domains are equal
+- their codomains are equal
+- $f(x)=g(x)$ for all $x\in \text{dom}(f)$
+
+### Function types
+
+An **injective function**, **injection**, or **one-to-one function** is a function that maps only one $y$-value to each $x$.
+
+$$\forall x_1,x_2\in\text{dom}(f), \text{ if } f(x_1)=f(x_2),x_1=x_2$$
+
+A **surjective function**, **surjection**, or **onto** is a function that has its codomain equal to its range.
+
+$$
+\forall y\in\text{cod}(f),\exists x\in\text{dom}(f), f(x)=y \\
+\text{rang}(f)=\text{cod}(f)
+$$