ece108: add composition

docs/1b/ece108.md

@@ -573,3 +573,28 @@ Compositions are commutative but not associative.
 
 - $h(gf)=(hg)f$
 - $hgf\neq hfg$
+- $f, g$ are injective $\implies$ $gf$ is injective
+- $f, g$ are surjective $\implies$ $gf$ is surjective
+- $gf$ is injective $\implies$ $f$ is injective
+- $gf$ is surjective $\implies$ $g$ is surjective
+
+The **identity function** is the function that returns its argument. Generally, a function composed with its inverse is the identity function.
+
+$$
+\begin{align*}
+I:X&\to X \\
+x&\mapsto x
+\end{align*}
+$$
+
+If $f: X\to Y$ is bijective:
+
+- the identity on $Y$ is $f(f^{-1}(y))$
+- the identity on $X$ is $f^{-1}(f(x))$
+
+If $f: X\to Y$ and $g: Y\to Z$ are bijective:
+
+- $gf$ exists and is invertible
+- $f^{-1}g^{-1}=(gf)^{-1}$ and exists
+
+