# Category:Composite Mappings

This category contains results about Composite Mappings.
Definitions specific to this category can be found in Definitions/Composite Mappings.

Let $S_1$, $S_2$ and $S_3$ be sets.

Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$.

### Definition 1

The composite mapping $f_2 \circ f_1$ is defined as:

$\forall x \in S_1: \map {\paren {f_2 \circ f_1} } x := \map {f_2} {\map {f_1} x}$

### Definition 2

The composite of $f_1$ and $f_2$ is defined and denoted as:

$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \tuple {\map {f_1} x, z} \in f_2}$

### Definition 3

The composite of $f_1$ and $f_2$ is defined and denoted as:

$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \map {f_1} x = y \land \map {f_2} y = z}$