# Category:Composite Mappings

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This category contains results about **Composite Mappings**.

Definitions specific to this category can be found in Definitions/Composite Mappings.

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:

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

## Subcategories

This category has the following 11 subcategories, out of 11 total.

### C

### E

### I

## Pages in category "Composite Mappings"

The following 67 pages are in this category, out of 67 total.

### C

- Composite Mapping is Mapping
- Composite of Bijections is Bijection
- Composite of Continuous Mappings at Point between Metric Spaces is Continuous at Point
- Composite of Continuous Mappings between Metric Spaces is Continuous
- Composite of Continuous Mappings between Normed Vector Spaces is Continuous
- Composite of Continuous Mappings is Continuous
- Composite of Continuous Mappings is Continuous/Corollary
- Composite of Continuous Mappings is Continuous/Point
- Composite of Epimorphisms is Epimorphism
- Composite of Homomorphisms is Homomorphism
- Composite of Homomorphisms is Homomorphism/Algebraic Structure
- Composite of Injection on Surjection is not necessarily Either
- Composite of Injections is Injection
- Composite of Inverse of Mapping with Mapping
- Composite of Mapping with Inverse
- Composite of Mapping with Inverse of Another is Identity implies Mappings are Equal
- Composite of Ordered Semigroup Isomorphisms is Isomorphism
- Composite of Quotient Mappings
- Composite of Surjection on Injection is not necessarily Either
- Composite of Surjections is Surjection
- Composite of Three Mappings in Cycle forming Injections and Surjection
- Composite with Constant Mapping is Constant Mapping
- Composition of 3 Mappings where Pairs of Mappings are Bijections
- Composition of Addition Mappings on Natural Numbers
- Composition of Cartesian Products of Mappings
- Composition of Commuting Idempotent Mappings is Idempotent
- Composition of Direct Image Mappings of Mappings
- Composition of Direct Image Mappings of Relations
- Composition of Idempotent Mappings
- Composition of Identification Mappings is Identification Mapping
- Composition of Inverse Image Mappings of Mappings
- Composition of Mapping with Inclusion is Restriction
- Composition of Mapping with Mapping Restricted to Image
- Composition of Mappings is Associative
- Composition of Mappings is Composition of Relations
- Composition of Mappings is Left Distributive over Homomorphism of Pointwise Operation
- Composition of Mappings is not Commutative
- Composition of Mappings is Right Distributive over Pointwise Operation
- Composition of Measurable Mappings is Measurable
- Composition of Product Mappings on Natural Numbers
- Composition of Repeated Compositions of Injections
- Composition of Right Inverse with Mapping is Idempotent
- Composition of Three Mappings which form Identity Mapping
- Condition for Composite Mapping on Left
- Condition for Composite Mapping on Right
- Continuity of Composite with Inclusion

### I

- Image of Composite Mapping
- Image of Composite Mapping/Corollary
- Image of Preimage of Subset under Surjection equals Subset
- Images of Elements under Repeated Composition of Injection form Equivalence Classes
- Injection if Composite is Injection
- Inverse of Composite
- Inverse of Composite Bijection
- Isomorphism between Algebraic Structures induces Isomorphism between Induced Structures

### M

### P

### S

- Set of all Self-Maps under Composition forms Monoid
- Set of all Self-Maps under Composition forms Semigroup
- Subset equals Preimage of Image iff Mapping is Injection
- Subset of Codomain is Superset of Image of Preimage
- Subset of Domain is Subset of Preimage of Image
- Surjection if Composite is Surjection