Search results
Results from the WOW.Com Content Network
In mathematics, cardinality describes a relationship between sets which compares their relative size. [1] For example, the sets = ... additional terms may apply.
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.
The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable. Each finite set is well-orderable, but does not have an aleph as its cardinality.
2. A probability measure on the algebra of all subsets of some set 3. A measure on the algebra of all subsets of a set, taking values 0 and 1 measurable cardinal A measurable cardinal is a cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. Most (but not all) authors add the condition ...
Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2. Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. ISBN 3-540-00384-3. Kanamori, Akihiro; Magidor, M. (1978).
The category < of sets of cardinality less than and all functions between them is closed under colimits of cardinality less than . κ {\displaystyle \kappa } is a regular ordinal (see below). Crudely speaking, this means that a regular cardinal is one that cannot be broken down into a small number of smaller parts.
In mathematics, an element (or member) of a set is any one of the distinct objects that belong to that set. For example, given a set called A containing the first four positive integers (= {,,,}), one could say that "3 is an element of A", expressed notationally as .
Examples of cardinal functions in algebra are: Index of a subgroup H of G is the number of cosets. Dimension of a vector space V over a field K is the cardinality of any Hamel basis of V. More generally, for a free module M over a ring R we define rank () as the cardinality of any basis of this module.