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Within data modelling, cardinality is the numerical relationship between rows of one table and rows in another. Common cardinalities include one-to-one , one-to-many , and many-to-many . Cardinality can be used to define data models as well as analyze entities within datasets.
In systems analysis, a one-to-many relationship is a type of cardinality that refers to the relationship between two entities (see also entity–relationship model). For example, take a car and an owner of the car. The car can only be owned by one owner at a time or not owned at all, and an owner could own zero, one, or multiple cars.
A country has only one capital city, and a capital city is the capital of only one country. (Not valid for some countries).. In systems analysis, a one-to-one relationship is a type of cardinality that refers to the relationship between two entities (see also entity–relationship model) A and B in which one element of A may only be linked to one element of B, and vice versa.
For example, think of A as Authors, and B as Books. An Author can write several Books, and a Book can be written by several Authors. In a relational database management system, such relationships are usually implemented by means of an associative table (also known as join table, junction table or cross-reference table), say, AB with two one-to-many relationships A → AB and B → AB.
The cardinality of the natural numbers is denoted aleph-null (), while the cardinality of the real numbers is denoted by "" (a lowercase fraktur script "c"), and is also referred to as the cardinality of the continuum.
A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities.
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This can be generalized as follows: the unique model of cardinality κ of a countable κ-categorical theory is saturated. However, the statement that every model has a saturated elementary extension is not provable in ZFC. In fact, this statement is equivalent to [citation needed] the existence of a proper class of cardinals κ such that κ <κ ...