---
date: '2026-06-05'
description: relevant to functions and operations on top of a set
id: preimages
modified: 2026-06-05 23:10:36 GMT-04:00
tags:
  - math
  - math/sets
title: preimages
created: '2026-06-05'
published: '2026-06-05'
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slug: thoughts/preimages
permalink: https://aarnphm.xyz/thoughts/preimages.md
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---
also known as a _inverse image_, of a subset $B$ uonder a function $f: X \to Y$ is the set of all elements in the domain $X$ that maps to elements in $B$

if $f$ maps set $X$ (domain) to set $Y$ (codomain), then $B \subseteq Y$ is <span class="marker marker-h2">always</span> a subset of $X$

| properties   | notation                                      |
| ------------ | --------------------------------------------- |
| intersection | $f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$ |
| union        | $f^{-1}(A\cup B) = f^{-1}(A) \cup f^{-1}(B)$  |
| complement   | $f^{-1}(B^{C}) = (f^{-1}(B))^{C}$             |

A distinction to be made between _image_ and _preimage_:

> <span class="marker marker-h2">image of a set</span> we go from _left to right_. For $A \subseteq X\;, f(A) = \{f(x) \mid x \in A\}$ is the set of outputs generated by a set of inputs <br/><br/> <span class="marker marker-h3">preimage of a set</span> we go from _right to left_. For $B \subseteq Y\;, f^{-1}(B) = \{x\in X \mid\;f(x) \in B\}$ is the set of inputs that generated a set of outputs