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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.
In mathematics, cardinality describes a relationship between sets which compares their relative size. [1] For example ... to cardinality: one which compares ...
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 ...
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 .
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.
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
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.
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