ece108: add bijective/composition

docs/1b/ece108.md

@@ -538,9 +538,38 @@ An **injective function**, **injection**, or **one-to-one function** is a functi
 
 $$\forall x_1,x_2\in\text{dom}(f), \text{ if } f(x_1)=f(x_2),x_1=x_2$$
 
-A **surjective function**, **surjection**, or **onto** is a function that has its codomain equal to its range.
+A **surjective function**, **surjection**, or **onto** is a function that has its codomain equal to its range. A surjection $g:Y\to X$ exists if and only if an injection $f:X\to Y$ exists.
 
 $$
 \forall y\in\text{cod}(f),\exists x\in\text{dom}(f), f(x)=y \\
 \text{rang}(f)=\text{cod}(f)
 $$
+
+A **bijective function** is both injective and surjective.
+
+An **inverse relation** swaps the domain, codomain, and ordered pairs.
+
+$$
+\begin{align*{
+R^{-1}:Y&\to X \\
+R(x)&\mapsto x
+$$
+
+A function is **invective** or **invertible** if and only if it is bijective. All inversions are also bijective.
+
+$$f^{-1^{-1}}=f$$
+
+A **composition** maps the codomain of one to the domain of another function only if the first is a subset ($Y_1\subseteq Y_2$).
+
+$$
+\begin{align*}
+f&:X\to Y_1,x\mapsto f(x) \\
+g&:Y_2\to Z,y\mapsto g(y) \\
+gf&: X\to Z,x\mapsto g(f(x))
+\end{align*}
+$$
+
+Compositions are commutative but not associative.
+
+- $h(gf)=(hg)f$
+- $hgf\neq hfg$