Search results
Results from the WOW.Com Content Network
A person must have their own birth certificate, it is specific to that person by its Id number. One-to-one (optional on one side) person ←→ driving license: 1: 0..1 or ? A person may have a driving license, it is specific to that person by its Id number. One-to-many: order ←→ line item: 1: 1..* or + An order contains at least one item ...
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 the other direction, the binary expansions of numbers in the half-open interval [,), viewed as sets of positions where the expansion is one, almost give a one-to-one mapping from subsets of a countable set (the set of positions in the expansions) to real numbers, but it fails to be one-to-one for numbers with terminating binary expansions ...
The continuum hypothesis says that =, i.e. is the smallest cardinal number bigger than , i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is independent of ZFC , a standard axiomatization of set theory; that is, it is impossible to prove the continuum ...
The following even stronger large cardinal properties are not consistent with the axiom of choice, but their existence has not yet been refuted in ZF alone (that is, without use of the axiom of choice). weakly Reinhardt cardinal, Reinhardt cardinal, Berkeley cardinal, super Reinhardt cardinal, totally Reinhardt cardinal
Cardinal numbers can be defined as follows. Define two sets to have the same size by: there exists a bijection between the two sets (a one-to-one correspondence between the elements). Then a cardinal number is, by definition, a class consisting of all sets of the same size. To have the same size is an equivalence relation, and the cardinal ...
In his controversial 1878 paper, Cantor explicitly defined the notion of "power" of sets and used it to prove that the set of all natural numbers and the set of all rational numbers are equinumerous (an example where a proper subset of an infinite set is equinumerous to the original set), and that the Cartesian product of even a ...
Cardinal functions are widely used in topology as a tool for describing various topological properties. [2] [3] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [4] prefer to define the cardinal functions listed below so that they never taken on finite cardinal numbers as values; this requires modifying some of the ...