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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 ...
Given two sets and , the set of binary relations between them (,) can be equipped with a ternary operation [,,] = where denotes the converse relation of . In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps , heaps, and generalized heaps.
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory , this is taken as the definition of "same number of elements" ( equinumerosity ), and generalising this definition to infinite sets leads to the concept of cardinal ...
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.
This implies trivially that (b, –a) is a linear relation between a and b. Therefore, given a generating set , …, of an ideal I, one calls trivial relation or trivial syzygy every element of the submodule the syzygy module that is generated by these trivial relations between two generating elements. More precisely, the module of trivial ...
For a ≤ b, the closed interval [a, b] is the set of elements x satisfying a ≤ x ≤ b (that is, a ≤ x and x ≤ b). It contains at least the elements a and b. Using the corresponding strict relation "<", the open interval (a, b) is the set of elements x satisfying a < x < b (i.e. a < x and x < b). An open interval may be empty even if a < b.
A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }. [5] Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets. A derived binary relation between two sets is the subset relation, also called set inclusion.
The equality equivalence relation is the finest equivalence relation on any set, while the universal relation, which relates all pairs of elements, is the coarsest. The relation "is finer than " on the collection of all equivalence relations on a fixed set is itself a partial order relation, which makes the collection a geometric lattice.