Reflexive relation

A bi­nary re­la­tion over some set is re­flex­ive when ev­ery el­e­ment of that set is re­lated to it­self. (In sym­bols, a re­la­tion \(R\) over a set \(X\) is re­flex­ive if \(\forall a \in X, aRa\).) For ex­am­ple, the re­la­tion \(\leq\) defined over the real num­bers is re­flex­ive, be­cause ev­ery num­ber is less than or equal to it­self.

A re­la­tion is anti-re­flex­ive when no el­e­ment of the set over which it is defined is re­lated to it­self.\(<) is an anti-re­flex­ive re­la­tion over the real num­bers. Note that a re­la­tion doesn’t have to be ei­ther re­flex­ive or anti-re­flex­ive; if Alice likes her­self but Bob doesn’t like him­self, then the re­la­tion “_ likes _” over the set \(\{Alice, Bob\}\) is nei­ther re­flex­ive nor anti-re­flex­ive.

The re­flex­ive clo­sure of a re­la­tion \(R\) is the union of \(R\) with the iden­tity re­la­tion; it is the small­est re­la­tion that is re­flex­ive and that con­tains \(R\) as a sub­set. For ex­am­ple, \(\leq\) is the re­flex­ive clo­sure of \(<).

Some other sim­ple prop­er­ties that can be pos­sessed by bi­nary re­la­tions are sym­me­try and tran­si­tivity.

A re­flex­ive re­la­tion that is also tran­si­tive is called a pre­order.

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