cardinal number

In set theory, the concept describing the number of elements contained in an arbitrary set. Also called potential. Two sets that can establish a one-to-one correspondence between elements are called reciprocal sets. For example, a set of three people and a set of three horses can establish a corresponding relationship, which is two equal sets. According to the equivalence relation, the sets are classified, and all mutually equivalent sets are classified into the same category. In this way, each set is classified into a certain category. The class to which any set A belongs is called the cardinality of set A, and it is recorded as (or | a |, or cardA). In this way, when a and b belong to the same class, the cardinality of a and b is the same, that is. When A and B belong to different classes, their cardinality is also different. Namely. If the cardinality of a single element set is recorded as 1, the cardinality of a set of two elements is recorded as 2, and so on, then the cardinality of any finite set is consistent with the natural number in the usual sense. Empty set? The cardinality of is also recorded as σ. Therefore, the cardinality of a finite set is also a "number" under the traditional concept. But for infinite sets, the traditional concept has no number, and according to the cardinality concept, infinite sets also have cardinality. For example, any countable set has the same cardinality as the natural number set n, that is, all countable sets are equal cardinality sets. Not only that, but also it can be proved that the cardinality of real number set r is different from countable set, that is. So the cardinality of a set is a generalization of the concept of number. Cardinality can compare sizes. Suppose that the cardinality of A and B are A and β respectively, that is, = A and = β. If a and a subset of b are equivalent, the cardinal number of A is not greater than that of B, and it is marked as a≤β, or β≥ A. If A is less than or equal to β, but A is not equal to β (that is, A is not equal to B), it is marked as A < β, or β > A, and the cardinal number can be operated. Let = a, = β, a ∩ b =, then the sum of A and β = a+β. Let = a, = β, A×B be the product set of A and B, and designate it as the product of A and β, and write it as = =a β.