Search results
Results from the WOW.Com Content Network
Binary relations over sets and can be represented algebraically by logical matrices indexed by and with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over and and a ...
It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations. The binary operations of set union and intersection satisfy many identities. Several of these identities or "laws" have well established names.
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. A binary relation between sets A and B is a subset of A × B. The (a, b) notation may be used for other purposes, most notably as denoting open intervals on the real number line ...
A logical matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets. It is an important tool in combinatorial mathematics and theoretical computer science.
[8] [9] This definition is equivalent to a partial order on a setoid, where equality is taken to be a defined equivalence relation rather than set equality. [10] Wallis defines a more general notion of a partial order relation as any homogeneous relation that is transitive and antisymmetric. This includes both reflexive and irreflexive partial ...
The above concept of relation [a] has been generalized to admit relations between members of two different sets (heterogeneous relation, like "lies on" between the set of all points and that of all lines in geometry), relations between three or more sets (finitary relation, like "person x lives in town y at time z "), and relations between ...
OCaml's standard library contains a Set module, which implements a functional set data structure using binary search trees. The GHC implementation of Haskell provides a Data.Set module, which implements immutable sets using binary search trees. [9] The Tcl Tcllib package provides a set module which implements a set data structure based upon TCL ...
For a binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold: each element of X must be paired with at least one element of Y, no element of X may be paired with more than one element of Y, each element of Y must be paired with at least one element of X, and