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.

The bit weight is a unit value corresponding to each fixed position in the exponential system.

For multi-digits, the numerical value represented by "1" on a certain digit is called bit weight. For example, the bit weight of the second decimal place is 10, and that of the third decimal place is100; In the binary system, the bit weight of the second bit is 2 and the bit weight of the third bit is 4. For N-ary numbers, the bit weight of the I-th bit in the integer part is n (I- 1), and the bit weight of the J-th bit in the decimal part is n-j. ..

The numerical value represented by a number is equal to the number itself multiplied by a constant related to its number of digits, which is called "bit weight" or "weight" for short.

Extended data

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.

Ordinary property

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.

Other attributes

The index has some interesting features:

| x | 0 = 1 (strangely, 0 0 = 1).

If y is not empty, 0^|Y| = 0.

1^|Y| = 1 .

|X| ≤ |Y| Then | X | | | z |≤| Y | | | z |

If |X| and |Y| are both finite sets and greater than 1, and z is an infinite set, then | x ||| z | = | y || z |.

If x is an infinite set and y is a non-empty finite set, then | x ||| y | = | x |.

Baidu Encyclopedia-Cardinality

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