# 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.

## 7.2.1. *Membership*

The first concept ...