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A bijective function, f: X → Y, from set X to set Y demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4. Aleph-null, the smallest infinite cardinal. In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set.
Two sets have the same cardinality if there exists a bijection (a.k.a., one-to-one correspondence) from to , [10] that is, a function from to that is both injective and surjective.
That at least one of and holds turns out to be equivalent to the axiom of choice. Nevertheless, most of the interesting results on cardinality and its arithmetic can be expressed merely with = c . The goal of a cardinal assignment is to assign to every set A a specific, unique set that is only dependent on the cardinality of A .
They were introduced by the mathematician Georg Cantor [1] and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ). [2] [a] The cardinality of the natural numbers is ℵ 0 (read aleph-nought, aleph-zero, or aleph-null), the next larger cardinality of a well-ordered set is aleph-one ℵ 1, then ℵ 2 and so on.
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. One owner could have many cars, one-to-many. In a relational database, a one-to-many relationship exists when one record is related to many records of another table. A one-to ...
The most frequently used cardinal function is the function that assigns to a set A its cardinality, denoted by |A|. Aleph numbers and beth numbers can both be seen as cardinal functions defined on ordinal numbers. Cardinal arithmetic operations are examples of functions from cardinal numbers (or pairs of them) to cardinal numbers.
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.
The simplest way to introduce cardinals is to add a primitive notion, Card(), and an axiom of cardinality to ZF set theory (without axiom of choice). [2] Axiom of cardinality: The sets A and B are equinumerous if and only if Card(A) = Card(B)