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.

Cardinal arithmetic:

We can define some arithmetic operations on cardinality, which is a generalization of natural number operations. Given set? X and y, define x+y = {(x, 0):X∈X }∨{(Y, 1): y ∈ y}, then the radix sum is |X|+|Y| = |X+Y|. If x and y do not intersect, |X|+|Y| = |X ∪ Y|. The radix product is | x |||| y | =| X × Y|, where x× y is the cartesian product of x and y, and the radix exponent is | x ||| y | = | x y |, where x y is the set of all functions from y to X.

Cardinality can compare sizes:

Let the cardinality of A and B be a and β respectively, that is |A|=a and |B|=β. If subsets of A and B are equivalent, the cardinal number of A is not greater than 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 and B are not equal), it is marked as A < β, or β >. Answer.

In the case of admitting axiom of choice, we can prove the disambiguation theorem of cardinality-the cardinality of any two sets can be compared, that is, there are no sets A and B, so that A cannot be equal to any subset of B, and B cannot be equal to any subset of A. ..