Composition of functions

     f          g
           1 - - - -> x
a -------> 2 -------> x
b -------> 3 -------> y
           4 - - - -> y
𝒜          𝓑         𝒞

f : A -> B is an injective function (1:1), so it doesn't cover the whole codomain (an injection that does is a bijection).

A	B	C	g ∘ f
_	1	x
a	2	x	a ⟼ x
b	3	y	b ⟼ y
_	4	y

g : B -> C is a surjection (onto), so it covers the while codomain but it is not 1:1, i.e. some elements in codomain are associated with more then one domain element, { 0 ⟼ w, 1 ⟼ w, 3 ⟼ y, 4 ⟼ y }.

However, the composition of f and g is bijective.

f : A -> B is injective, g : B -> C is surjective, but their composition g ∘ f : A -> C is bijective. Composition ignores some mappings from the common (middle) set B. Namely, it ignores { 1 ⟼ x, 4 ⟼ y }, i.e. these 2 mappings are excluded.

Composition

        ----f---> x

p ------> a ----g---> y ----h---> z

 f          g

a -------> 1 ------> x b -------> 2 ------> y c -------> 3 ------> y

 f          g

a -------> 1 ------> x b -------> 2 ------> y c -------> 3 ------> y

Composability

Composition of functions is possible when the codomain of one function is the same as the domain of another, cod(f) = dom(g) => ∃h. h = g ∘ f

      g ∘ f
a ------------> c
|               .
|               .
| f             . 1ᴄ
|               .
↓       g       .
b ------------> c


     f         g
a ------> b ------> c
a ----------------> c
        g ∘ f

Associativity

Composition is associative: h ∘ (g ∘ f) = (h ∘ g) ∘ f = h ∘ g ∘ f

```dot {engine="dot"} digraph { D [label="d"] A [label="a"] C [label="c"] B [label="b"] A -> B [label="f", constraint=false] B -> C [label="g", constraint=true] C -> D [label="h", constraint=false] A -> C [label="g.f", constraint=false] A -> D [label="h.g.f", constraint=true] }

```