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Let (,) and (,) be ordered pairs. Then the characteristic (or defining) property of the ordered pair is: (,) = (,) = =.. The set of all ordered pairs whose first entry is in some set A and whose second entry is in some set B is called the Cartesian product of A and B, and written A × B.
In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. [1] As an example, "is less than" is a relation on the set of natural numbers; it holds, for instance, between the values 1 and 3 (denoted as 1 < 3), and likewise between 3 and 4 (denoted as 3 < 4), but not between the values 3 ...
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.
Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be
A partially ordered set (poset for short) is an ordered pair = (,) consisting of a set (called the ground set of ) and a partial order on . When the meaning is clear from context and there is no ambiguity about the partial order, the set X {\displaystyle X} itself is sometimes called a poset.
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:
A nested set collection or nested set family is a collection of sets that consists of chains of subsets forming a hierarchical structure, like Russian dolls. It is used as reference concept in scientific hierarchy definitions, and many technical approaches, like the tree in computational data structures or nested set model of relational databases .
The set of integers and the set of even integers have the same order type, because the mapping is a bijection that preserves the order. But the set of integers and the set of rational numbers (with the standard ordering) do not have the same order type, because even though the sets are of the same size (they are both countably infinite), there ...