# cardinality

## English

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### Etymology

cardinal +‎ -ity.

### Pronunciation

•  Audio (US): (file)

### Noun

cardinality (plural cardinalities)

1. (set theory, of a set) The number of elements a given set contains.
Synonym: power
The empty set has a cardinality of zero.
• 2005, Johan de Jong, “Set Theory”, in The Stacks Project[1], retrieved 2018-2-26:
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 ${\displaystyle \alpha =|A|}$ to indicate this.
• 2006, Michael Smithson, Jay Verkuilen, Fuzzy Set Theory: Applications in the Social Sciences, SAGE Publications, page 37:
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.
• 2012, Adolf Grünbaum, Robert S. Cohen, Marx W. Wartofsky, Philosophical Problems of Space and Time, 2nd edition, Springer, page 487:
Clearly, in this example, the sensitivity to the cardinalities takes the weaker form ${\displaystyle F[M(A)]={\overset {=}{A}}}$ of a single-valued function from the measure to the cardinality rather than the stronger form ${\displaystyle M(A)=f({\overset {=}{A}})}$ of a function from the cardinality to the measure.
2. The number of terms that can inhabit a type; the possible values of a type.
• 2021, Martin Odersky et al., chapter 19, in Programming in Scala, 5th edition, Artima, →ISBN:
For many types, such as String, the set of possible values is unlimited. Such types have an infinite cardinality.
3. The property of a relationship between a database table and another one, specifying whether it is one-to-one, one-to-many, many-to-one, or many-to-many.
4. (religion) The status of being cardinalitial

#### Usage notes

(set theory): The cardinality of an infinite set is an infinite cardinal number. The smallest such number, called aleph-null and denoted ℵ₀, describes the natural numbers; the next is aleph-one. The cardinality of the real numbers is greater than aleph-null, though whether it is equal to aleph-one is the subject of the continuum hypothesis.