7 Basic Reminders in Set Theory

This chapter aims to briefly summarize the basic notions of Set Theory that should be necessary for presenting the mathematical imaging frameworks for both gray-tone and binary images.

7.1. Mathematical disciplines

The mathematical discipline of reference is the theory of sets, or for short Set Theory [RUB 67, DEV 93, BOU 04c]. The notion of set is fundamental in modern Mathematics because most of the mathematical entities (numbers, relations, operations, functions, etc.) are defined in terms of sets.

7.2. Sets and elements

A set, denoted as is a collection of well-defined and distinct elements, denoted A subset of a set is a subcollection of well-defined and distinct elements of . A singleton is also a set with exactly one element. For example, the set is a singleton related to the element The empty set is denoted by Ø.