One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge's two vertices being the two entities so related.
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In mathematics, an "'equivalence relation "'is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation.
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For relations, a symmetric relation is analogous to a commutative operation, in that if a relation " R " is symmetric, then a R b \ Leftrightarrow b R a.
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A binary relation R over processes is a " barbed bisimulation " if it is a symmetric relation which satisfies that for every pair of elements ( p, q ) \ in R we have that
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He explains that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, and notes that Casimir went on to derive anti-symmetric relations concerning variables which are odd functions of the velocities of the molecules.
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Reachability is not a symmetric relation since, by definition, no point may be reachable from a non-core point, regardless of distance ( so a non-core point may be reachable, but nothing can be reached from it ).
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In terms of the homomorphism problem, every such problem is equivalent to the existence of a homomorphism from a relational structure to a given fixed undirected graph ( an undirected graph can be regarded as a relational structure with a single binary symmetric relation ).
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Note that " parent " is not a symmetric relation ( because if I'm your parent-then you can't be my parent ), nor is " offspring "; However " spouse " is a symmetric natural relation ( because if I'm your spouse-then you too must be my spouse ).
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Take as the vertices of a graph all the prime numbers that are congruent to 1 modulo 4, and connect two vertices by an edge whenever one of the two numbers is a quadratic residue modulo the other ( by quadratic reciprocity and the restriction of the vertices to primes congruent to 1 mod 4, this is a symmetric relation, so it defines an undirected graph ).
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I'm looking for additional languages, other than the Slavic ones, which do not have the property English has, i . e either : languages that don't have a word denoting a symmetric relation which is not a symmetric " natural " relation, or languages that have a word denoting another symmetric relation ( i . e . other than " sibling in law " ) which is not a symmetric " natural " relation.
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