# Category:Preimages under Mappings

This category contains results about **Preimages under Mappings**.

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

Let $f^{-1} \subseteq T \times S$ be the inverse of $f$, considered as a relation:

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

### Preimage of Element

Every $s \in S$ such that $\map f s = t$ is called **a preimage** of $t$.

**The preimage** of an element $t \in T$ is defined as:

- $\map {f^{-1} } t := \set {s \in S: \map f s = t}$

This can also be expressed as:

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

That is, the **preimage of $t$ under $f$** is the image of $t$ under $f^{-1}$.

### Preimage of Subset

Let $Y \subseteq T$.

The **preimage of $Y$ under $f$** is defined as:

- $f^{-1} \sqbrk Y := \set {s \in S: \exists t \in Y: \map f s = t}$

That is, the **preimage of $Y$ under $f$** is the image of $Y$ under $f^{-1}$, where $f^{-1}$ can be considered as a relation.

### Preimage of Mapping

The **preimage of $f$** is defined as:

- $\Preimg f := \set {s \in S: \exists t \in T: f \paren s = t}$

That is:

- $\Preimg f := f^{-1} \sqbrk T$

where $f^{-1} \sqbrk T$ is the image of $T$ under $f^{-1}$.

In this context, $f^{-1} \subseteq T \times S$ is the the inverse of $f$.

## Subcategories

This category has the following 5 subcategories, out of 5 total.

## Pages in category "Preimages under Mappings"

The following 22 pages are in this category, out of 22 total.

### I

### P

- Preimage of Cover is Cover
- Preimage of Horizontal Section of Function is Horizontal Section of Preimage
- Preimage of Image of Subset under Injection equals Subset
- Preimage of Intersection under Mapping
- Preimage of Mapping equals Domain
- Preimage of Set Difference under Mapping
- Preimage of Set Difference under Mapping/Corollary 1
- Preimage of Subset is Subset of Preimage
- Preimage of Subset under Composite Mapping
- Preimage of Subset under Mapping equals Union of Preimages of Elements
- Preimage of Union Mapping is Union of Preimages
- Preimage of Union under Mapping
- Preimage of Vertical Section of Function is Vertical Section of Preimage