# 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:

- $f^{-1} = \left\{{\left({t, s}\right): f \left({s}\right) = t}\right\}$

### 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.

If no element of $Y$ has a **preimage**, then $f^{-1} \sqbrk Y = \O$.

### 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 3 subcategories, out of 3 total.

## Pages in category "Preimages under Mappings"

The following 23 pages are in this category, out of 23 total.

### I

### P

- Preimage of Cover is Cover
- 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 under Mapping
- Preimage of Union under Mapping/Family of Sets
- Preimage of Union under Mapping/Family of Sets/Proof 1
- Preimage of Union under Mapping/Family of Sets/Proof 2
- Preimage of Union under Mapping/General Result