Search results
Results from the WOW.Com Content Network
Since set theorists work with infinite sets of arbitrarily large cardinalities, the default definition among this group of mathematicians of an enumeration of a set tends to be any arbitrary α-sequence exactly listing all of its elements. Indeed, in Jech's book, which is a common reference for set theorists, an enumeration is defined to be ...
A set of polygons in an Euler diagram This set equals the one depicted above since both have the very same elements.. In mathematics, a set is a collection of different [1] things; [2] [3] [4] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other ...
To effectively convert a Lehmer code d n, d n−1, ..., d 2, d 1 into a permutation of an ordered set S, one can start with a list of the elements of S in increasing order, and for i increasing from 1 to n set σ i to the element in the list that is preceded by d n+1−i other ones, and remove that element from the list.
Total orders, orderings that specify, for every two distinct elements, which one is less than the other; Weak orders, generalizations of total orders allowing ties (represented either as equivalences or, in strict weak orders, as transitive incomparabilities) Well-orders, total orders in which every non-empty subset has a least element
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering.
In a partially ordered set there may be some elements that play a special role. The most basic example is given by the least element of a poset. For example, 1 is the least element of the positive integers and the empty set is the least set under the subset order. Formally, an element m is a least element if: m ≤ a, for all elements a of the ...
The algorithm starts at the beginning of the data set. It compares the first two elements, and if the first is greater than the second, it swaps them. It continues doing this for each pair of adjacent elements to the end of the data set. It then starts again with the first two elements, repeating until no swaps have occurred on the last pass. [34]
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.