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Reflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself. In other words, a relation ~ on a set ''S'' is reflexive when ''x'' ~ ''x'' holds true for every ''x'' in ''S'', formally: when ŌłĆ''x''Ōłł''S'': ''x''~''x'' holds.ŃĆöLevy 1979:74ŃĆĢŃĆöRelational Mathematics, 2010ŃĆĢ An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity.
==Related terms==
A relation that is , or anti-reflexive, is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y) on the real numbers. Note that not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not (i.e., neither all nor none are). For example, the binary relation "the product of ''x'' and ''y'' is even" is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.
A relation ~ on a set ''S'' is called quasi-reflexive if every element that is related to some element is also related to itself, formally: if ŌłĆ''x'',''y''Ōłł''S'': ''x''~''y'' ŌćÆ ''x''~''x'' Ōł¦ ''y''~''y''. An example is the relation "has the same limit as" on the set of sequences of real numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself.
The reflexive closure Ōēā of a binary relation ~ on a set ''S'' is the smallest reflexive relation on ''S'' that is a superset of ~. Equivalently, it is the union of ~ and the identity relation on ''S'', formally: (Ōēā) = (~) Ōł¬ (=). For example, the reflexive closure of ''x''<''y'' is ''x''Ōēż''y''.
The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set ''S'' is the smallest relation Ōēå such that Ōēå shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalent to the complement of the identity relation on ''S'' with regard to ~, formally: (Ōēå) = (~) \ (=). That is, it is equivalent to ~ except for where ''x''~''x'' is true. For example, the reflexive reduction of ''x''Ōēż''y'' is ''x''<''y''.