# Classification of relations

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Binary relations can be classified in many ways. Some is described in the list below. In the examples a and b are, say, real numbers, when using mathematical symbols.

Type of relation Condition Description Examples Reflexive
reflexive aRa for every a every element relates to itself a=b, as old as
antireflexive not aRa for any a no elements relates to itself a>b; is the mother of
transitive aRb and bRc implies aRc the relation form “chains” like a>b>c  where a>c a>b , a=b
cotransitive aRb and cRb implies aRc if two relates to a third, then they relate to each other a=b,  lives in the same country as
symmetric aRb implies bRa if a relates to b then b relates to a a=b, “is a relative to”
antisymmetric aRb and bRa implies a=b if a and b relates to each other then they are the same $a\geq b$
asymmetric  aRb implies not bRa  if a relates to b then b does not relate to a $a>b$
total aRb or bRa every pair of elements relates to each other in one way or the other $a\geq b$ Up a level : Algebra and Arithmetic
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