
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 ∀''x''∈''S'': ''x''~''x'' holds.〔Levy 1979:74〕〔Relational Mathematics, 2010〕 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 antireflexive, 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 quasireflexive if every element that is related to some element is also related to itself, formally: if ∀''x'',''y''∈''S'': ''x''~''y'' ⇒ ''x''~''x'' ∧ ''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 ≃ 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: (≃) = (~) ∪ (=). For example, the reflexive closure of ''x''<''y'' is ''x''≤''y''. The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set ''S'' is the smallest relation ≆ such that ≆ 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: (≆) = (~) \ (=). That is, it is equivalent to ~ except for where ''x''~''x'' is true. For example, the reflexive reduction of ''x''≤''y'' is ''x''<''y''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Reflexive relation」の詳細全文を読む スポンサード リンク
