Mathematically, cardinality is a concept describing the size of any set in set theory. 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 one-to-one correspondence, which is two equal sets.

Indicates the quantity of the order. There are many ways to express ordinal numbers in Chinese. Usually add "first" before integers, such as: first and second. There are also those that only use cardinality. Such as: five elements: water, fire, wood, gold and earth. In addition, there are some established expressions, such as: first time, last time, first time, first month, eldest daughter, youngest son.

When the extended data is in a finite set, these operations are the same as natural numbers. Generally speaking, they also have the characteristics of ordinary arithmetic operations:

Addition and multiplication are interchangeable, that is, |X|+|Y|=|Y|+|X| and | x | | y | = | y ||| x |.

Addition and multiplication conform to the law of association, (|X|+|Y|)+|Z|=|X|+(|Y|+|Z|) and (| x ||| y |) | z | = | x | (| y | | z |).

Distribution law, that is, (| x |+| y|) | z | = | x|||| z |+| y|||| z | | = | x ||||||| | z | |

The addition and multiplication of infinite sets (assuming axiom of choice) are very simple. If both x and y are not empty, and one of them is an infinite set, then | x |+| y | = | x | | = max {| x |, | y |}.

Remember that 2 | x | is the radix of the power set of X. From the diagonal argument, we can know that 2 | x | > | X | means that there is no maximum radix. In fact, cardinality classes are real classes.

References:

Baidu encyclopedia? cardinal number

Baidu encyclopedia? ordinal