Cardinality

A set with the cardinality of the natural numbers is called "countably infinite".

The empty set has a cardinality of zero.

The cardinality of a set A is the least ordinal α such that there exists a bijection between A and α. We sometimes use the notation #92;alpha#61;#124;A#124; to indicate this.

For fuzzy sets, the concept of set size or cardinality is both richer and more problematic than it is for crisp sets. It is richer because, as we shall see, we may use more than one kind of cardinality.

Clearly, in this example, the sensitivity to the cardinalities takes the weaker form F#91;M(A)#93;#61;#92;overset#123;#61;#125;#123;A#125; of a single-valued function from the measure to the cardinality rather than the stronger form M(A)#61;f(#92;overset#123;#61;#125;#123;A#125;) of a function from the cardinality to the measure.

For many types, such as String, the set of possible values is unlimited. Such types have an infinite cardinality.