Advanced Mathematics

3.5Image of a Set

Purpose of Section

To introduce the concept of the image and inverse image of a set. We show that unions of sets are preserved under a mapping, whereas intersections are preserved under one‐to‐one functions. On the other hand, the inverse function preserves both unions and intersections.

3.5.1 Introduction

In many areas of mathematics, be it topology, measure theory, real analysis, and others, one seeks to find the image of a set A under the action of a function f : A → B, yielding the image set

In medical imaging, such as CT scans, MRI imaging, X‐rays, one is not interested in images of points, but of images of sets. Although the image viewed by the medical professionals is not the kind of function we have studied thus far, it is a function nevertheless.

Medical imagings are images of sets under some function.

This discussion motivates the image and inverse image of a set.

Definition Image and Inverse Image of a Set

Given sets X, Y let f : X → Y and A ⊆ X. As x takes on all values in the set A ⊆ X, the set of values f(x) ⊆ Y defines the image of A:

Also, for B ⊆ Y we can define the inverse image of B as the set

Note that f⁻¹(B) is a well‐defined ...