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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.
This means that the bisimilarity relation ∼ is the union of all bisimulations: (,) precisely when (,) for some bisimulation R. The set of bisimulations is closed under union; [Note 1] therefore, the bisimilarity relation is itself a bisimulation. Since it is the union of all bisimulations, it is the unique largest bisimulation.
Orders are special binary relations. Suppose that P is a set and that ≤ is a relation on P ('relation on a set' is taken to mean 'relation amongst its inhabitants', i.e. ≤ is a subset of the cartesian product P x P). Then ≤ is a partial order if it is reflexive, antisymmetric, and transitive, that is, if for all a, b and c in P, we have that:
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation R ; S from two given binary relations R and S. In the calculus of relations, the composition of relations is called relative multiplication, [1] and its result is called a relative product.
A homogeneous relation over a set is a binary relation over and itself, i.e. it is a subset of the Cartesian product . [14] [32] [33] It is also simply called a (binary) relation over .
Pages in category "Properties of binary relations" The following 22 pages are in this category, out of 22 total. This list may not reflect recent changes. A.
A relation is reflexive if, and only if, its complement is irreflexive. A relation is strongly connected if, and only if, it is connected and reflexive. A relation is equal to its converse if, and only if, it is symmetric. A relation is connected if, and only if, its complement is anti-symmetric.
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