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When two sets, and , have the same cardinality, it is usually written as | | = | |; however, if referring to the cardinal number of an individual set , it is simply denoted | |, with a vertical bar on each side; [3] this is the same notation as absolute value, and the meaning depends on context.
The new cardinal number of the set of real numbers is called the cardinality of the continuum and Cantor used the symbol for it. Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number ( ℵ 0 {\displaystyle \aleph _{0}} , aleph-null), and that for every ...
The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable. Each finite set is well-orderable, but does not have an aleph as its cardinality.
Standard set theory symbols with their usual meanings (is a member of, equals, is a subset of, is a superset of, is a proper superset of, is a proper subset of, union, intersection, empty set) ∧ ∨ → ↔ ¬ ∀ ∃ Standard logical symbols with their usual meanings (and, or, implies, is equivalent to, not, for all, there exists) ≡
The category < of sets of cardinality less than and all functions between them is closed under colimits of cardinality less than . κ {\displaystyle \kappa } is a regular ordinal (see below). Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.
so that the second beth number is equal to , the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number is the cardinality of the power set of the continuum.
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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.