# 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] }

\`\`\`