Symmetric Relation

A Symmetric Relation is a relation that is invariant to permutations of its relation arguments.

• Context:
  • More formally:
    • IF a,b in S and R(a,b) is True THEN R(b,a) is True.
    • ∀a₁,...,a_n ∈ S: R(a₁,...,a_n) → R(a_j,...,a_k), where [math]\displaystyle{ j }[/math] and [math]\displaystyle{ k }[/math] in [1,n].
• Example(s):
  • If R(a,b) and R(b,b) are true and R(a,c) is false then R(b,a) is true and R(c,a) is false. The relations of R(a,a), R(b,c) and R(c,c) however are undefined.
  • Equality Relation: IsEqualTo(43.7,43.7) clearly remains true if you swap the two values.
  • IsSiblingTo(A,B): If IsSiblingTo(Jim Doe,Joe Doe) is TRUE Then IsSiblingTo(Joe Doe,Jim Doe) is also TRUE
  • Cannot also be both a Transitive Relation and also an Irreflexive Relation.
  • http://www.gabormelli.com/images/SymmetricRelationFig1.gif Note how if an edge is TRUE in one direction then is also TRUE in the other direction.
• Counter-Example(s):
  • an Antisymmetric Relation.
  • IsParentTo(A,B): If IsParentOf(Ann Doe,Joe Doe) is TRUE Then IsParentOf(Joe Doe,Ann Doe) is NOT also TRUE
  • GreaterThanOrEqualTo(X,Y)
  • Proper Subset(X,Y).
• See: Reflexive Relation, Transitive Relation, Symmetric Function, Symmetric Relationship.

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References

2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Symmetry_in_mathematics#Symmetric_functions
  • In the case of symmetric functions, the value of the output is invariant under permutations of variables. From the form of an equation one may observe that certain permutations of the unknowns result in an equivalent equation. In that case the set of solutions is invariant under any permutation of the unknowns in the group generated by the aforementioned permutations. For example
• http://www.cs.odu.edu/~toida/nerzic/content/relation/property/property.html
  • Definition(symmetric relation): A relation [math]\displaystyle{ R }[/math] on a set A is called symmetric if and only if for any a, and b in A, whenever &lta, b> R, &ltb, a> [math]\displaystyle{ R }[/math].
  • Example 5: The relation = on the set of integers {1, 2, 3} is {<1, 1>, <2, 2> <3, 3> } and it is symmetric. Similarly = on any set of numbers is symmetric. However, < (or >), (or on any set of numbers is not symmetric.
  • Example 6: The relation "being acquainted with" on a set of people is symmetric.

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