Partitions and Relations

Abstract

In this chapter the foundations for models of subsequent chapters are discussed. S4 contains some definitions and first properties of fuzzy sets. The important device of mathematical embedding first appears here. Interestingly enough, however, a property of the embedded structure (that a subset and its complement reproduce the original set under union) which is not preserved under embedment turns out to be one of its most useful aspects! In S5 partitions of finite data sets are given a matrix-theoretic characterization, culminating in a theorem which highlights the difference between conventional and fuzzy partitions. S6 is devoted to exploration of the algebraic and geometric nature of the partition embedment: the main results are that fuzzy partition space is compact, convex, and has dimension n (c − 1). S7 considers hard and fuzzy relations in finite data sets and records some connections between relations and partitions that tie the observational and relation-theoretic approaches together.