Set-theoretic relations for metasets

ABSTRACT

The paper introduces partial set-theoretic relations for metasets and investigates their properties. Metaset is a concept of imprecise set designed to represent vague notions in Artificial Intelligence, whose idea stems from Boolean-valued techniques in classical set theory. The basic set relations are extended to functions valued in a Boolean algebra or unit interval. Important classical set theory axioms: extensionality and comprehension are formulated for metasets. The latter enables formal definitions of collections with blurred boundaries by using set-theory formulae. This facilitates representing and reasoning about imprecise data.

KEYWORDS:

• Metaset
• partial membership
• vagueness
• imprecise set
• set-theoretic relation
• knowledge representation

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1. Also known as Axiom Schema of Separation or Axiom Schema of Specification (Jech, 2006; Kunen, 1980).

2. $f_A(x)$ is the membership function of the fuzzy set $A$.

3. A crisp set is the common term used when referring to classical sets with sharp bounds.

4. See, Kunen (1980, Ch. VII, §2) for the justification of such type of induction.

5. This idea may be made rigorous by formalising logic within the set theory using Gödelisation (Kunen, 1980).

6. A set is hereditarily finite, when the transitive closure of set membership relation for this set is finite, i.e., the set itself is finite and its members are hereditarily finite.