# Category:Inverse Mappings

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

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

Let $f: S \to T$ be a mapping.

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$:

- $f^{-1} := \set {\tuple {t, s}: \map f s = t}$

Let $f^{-1}$ itself be a mapping:

- $\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$

and

- $\forall y \in T: \exists x \in S: \tuple {y, x} \in f$

Then $f^{-1}$ is called the **inverse mapping of $f$**.

## Subcategories

This category has the following 9 subcategories, out of 9 total.

## Pages in category "Inverse Mappings"

The following 39 pages are in this category, out of 39 total.

### B

### C

- Composite of Bijection with Inverse is Identity Mapping
- 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
- Continuous Linear Operator over Finite Dimensional Vector Space is Invertible
- Continuous Linear Operator over Infinite Dimensional Vector Space is not necessarily Invertible

### E

### I

- Image of Element under Inverse Mapping
- Image of Inverse Image
- Inverse Element of Bijection
- Inverse Element of Injection
- Inverse Mapping is Bijection
- Inverse Mapping is Unique
- Inverse of Bijection is Bijection
- Inverse of Composite Bijection
- Inverse of Identity Mapping
- Inverse of Injection is One-to-One Relation
- Inverse of Inverse of Bijection
- Inverse of Mapping is One-to-Many Relation
- Inverse of Mapping is Right-Total Relation
- Inverse of Ordered Semigroup Isomorphism is Isomorphism
- Inverse of Surjection is Relation both Left-Total and Right-Total
- Invertibility of Identity Transformation Plus Product of Two Continuous Linear Transformations
- Invertible Continuous Linear Operator has Unique Inverse