A Modern Introduction to Fuzzy Mathematics

Preface

In mathematics, we investigate the properties of abstract objects (e.g. numbers, geometric shapes, spaces) and the possible relationships between these objects. One basic characteristic of mathematics is that all these properties and relationships are absolute. Thus, properties and relationships are either true or false. Nothing else is meaningful. In other words, in mathematics there is no room for vagueness, for randomness, and for extremely small quantities. By introducing one of these qualities into mathematics, one can create alternative mathematics [1]. But how do we introduce vagueness into mathematics? One very simple way to achieve this is to allow notions like “small,” “large,” and “few.” However, another way is to modify the most basic object of mathematics, that is, to modify sets. In this respect, fuzzy mathematics is a form of alternative mathematics since it is based on a generalization of set membership. Simply put, in fuzzy mathematics, an element may belong to a degree to a set, while in ordinary mathematics, it either belongs or does not belong to a set. This simple idea has been applied to most fields of mathematics and so we can talk about fuzzy mathematics.

Even today, many researchers and thinkers consider fuzzy mathematics as a tool that can be used instead of probability theory to reason about or to work with a specific system. This text is based on the idea that vagueness is a basic notion and thus tries to present fuzzy mathematics as a form ...