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In computer science, a bidirectional map is an associative data structure in which the (,) pairs form a one-to-one correspondence. Thus the binary relation is functional in each direction: each v a l u e {\displaystyle value} can also be mapped to a unique k e y {\displaystyle key} .
(A minor modification needs to be made to the concept of the ordered triple (,,), as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.) [31] With this definition one can for instance define a binary relation over every set and its power set.
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms. A morphism (or arrow) R : A → B in this category is a relation between the sets A and B, so R ⊆ A × B. The composition of two relations R: A → B and S: B → C is given by (a, c) ∈ S o R ⇔ for some b ∈ B, (a, b) ∈ R and (b, c) ∈ ...
is a binary relation from consistent sets to elements of . The relation C ⊢ e {\displaystyle C\vdash e} , for C ∈ C {\displaystyle C\in {\mathcal {C}}} and e ∈ E {\displaystyle e\in E} is interpreted as meaning that when the events so far form set C {\displaystyle C} , this enables e {\displaystyle e} to be the next event.
A formal context is a triple K = (G, M, I), where G is a set of objects, M is a set of attributes, and I ⊆ G × M is a binary relation called incidence that expresses which objects have which attributes. [4] For subsets A ⊆ G of objects and subsets B ⊆ M of attributes, one defines two derivation operators as follows:
A Kripke structure is a variation of the transition system, originally proposed by Saul Kripke, [1] used in model checking [2] to represent the behavior of a system. It consists of a graph whose nodes represent the reachable states of the system and whose edges represent state transitions, together with a labelling function which maps each node ...
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.
Suppose X and Y are arbitrary sets and a binary relation R over X and Y is given. For any subset M of X, we define F(M ) = { y ∈ Y | mRy ∀m ∈ M }. Similarly, for any subset N of Y, define G(N ) = { x ∈ X | xRn ∀n ∈ N }. Then F and G yield an antitone Galois connection between the power sets of X and Y, both ordered by inclusion ⊆. [7]