Cardinalities of dense sets

Comparing Cardinalities of Sets

Students often have motivational difficulty in accepting the definitions of comparing cardinalities of sets. The key to the application of the concepts lies in the so called SchröderBernstein theorem, the proof of which is often avoided due to its difficulty level. In this short article, we discuss the remarkable results around this theorem and finally give a very simple proof. On the way, we a...

Cardinalities of fuzzy sets with triangular norms

This paper presents a variety of cardinality concepts offered by cardinality theory of triangular norm-based fuzzy sets, including the scalar and the "fuzzy" streams.

Order statistics and estimating cardinalities of massive data sets

We introduce a new class of algorithms to estimate the cardinality of very large multisets using constant memory and doing only one pass on the data. It is based on order statistics rather that on bit patterns in binary representations of numbers. We analyse three families of estimators. They attain a standard error of 1 √ M using M units of storage, which places them in the same class as the b...

Cardinalities of Fuzzy Sets and Fuzzy Quantifiers over Residuated Lattices

Cardinalities of k-distance sets in Minkowski spaces

A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1) points, with equality iff the unit ball is a parallelotope. We solve this conjecture in the affirmative for all 2-dimensional spaces and for spaces where the unit ball is...