PreorderY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Equivalence relationY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Disjoint unionIn mathematics, the disjoint union (or discriminated union) A⊔B{\displaystyle A\sqcup B} of the sets A and B is the set formed from the elements of A and B labelled (indexed) with the name of the set from which they come. So, an element belonging to both A and B appears twice in the disjoint union, with two different labels.Transitive closureY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Reflexive closureIn mathematics, the reflexive closure of a binary relationR{\displaystyle R} on a setX{\displaystyle X} is the smallest reflexive relation on X{\displaystyle X} that contains R{\displaystyle R}, i.e. the set R∪{(x,x)∣x∈X}{\displaystyle R\cup \{(x,x)\mid x\in X\}}.
Relation (mathematics)In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold. 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 and 1 nor between 4 and 4, that is, 3 < 1 and 4 < 4 both evaluate to false.
Transition systemIn theoretical computer science, a transition system is a state machine that may have infinite states. It is used to describe the potential behavior of discrete systems. It consists of states and transitions between states, which may be labeled with labels chosen from a set; the same label may appear on more than one transition. If the label set is a singleton, the system is essentially unlabeled, and a simpler definition that omits the labels is possible.
Composition of relationsIn the mathematics of binary relations, the composition of relations is the forming of a new binary relation R ; S {\displaystyle R\mathbin {;} S} from two given binary relations R {\displaystyle R} and S {\displaystyle S} . In the calculus of relations, the composition of relations is called relative multiplication, and its result is called a relative product. Function composition is the special case of composition of relations where all relations involved are functions.
Theoretical computer scienceTheoretical computer science is a subfield of computer science and mathematics that focuses on the abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely.BisimulationIn theoretical computer science, a bisimulation is a binary relation between state transition systems, associating systems that behave in the same way in that one system simulates the other and vice versa. Intuitively two systems are bisimilar if they, assuming we view them as playing a game according to some rules, match each other's moves. In this sense, each of the systems cannot be distinguished from the other by an observer.*As an Amazon Associate I earn from qualifying purchases.