Composition

algebra

Definition️

Composition

Given a function $f:X\to Y$ and a function $g:Y\to Z$, then $h:=g\circ f$ maps $X\to Z$. The result of a function composition is again a function.

$$(g\circ f)(x)=g(f(x))$$

Properties

Identity

Associativity

$$f\circ (g\circ h)=(f\circ g)\circ h$$

Injectivity

$$g\circ f\text{injective}⟺g\text{injective}\wedge f\text{injective}$$

Surjectivity

$$g\circ f\text{surjective}⟺g\text{surjective}\wedge f\text{surjective}$$