A Modern Introduction to Fuzzy Mathematics

2Fuzzy Sets and Their Operations

Based on the assumption that all mathematics can be built up from sets, one has to first define fuzzy sets and their variations. However, one needs to be familiar with certain notions and ideas in order to fully understand the notions that will be introduced. Thus, we start by exploring truth values and their algebras and continue with ‐norms and ‐conorms.

2.1 Algebras of Truth Values

Aristotle ('Aριστοτἑλης) was the first thinker who made a systematic study of logic in his Organon (Ὄργανον), a collection of six works on logic. However, the transformation of logic into something that looks like an algebra was started by the German philosopher and mathematician Gottfried Wilhelm Leibniz, who is primarily known as one of the two people who devised the differential calculus. Later on, George Boole in his The Mathematical Analysis of Logic [35] introduced in a way the truth values (true) and (false) and algebraic operations between them. This work was important because it showed that truth values can be represented by numbers and the various logical ...