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  text: 'A binary [3nt 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.\n\nA 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.\n\nThe **reflexive closure** of a relation $R$ is the union of $R$ with the [Identity_relation 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 $<$.\n\nSome other simple properties that can be possessed by binary relations are [Symmetric_relation symmetry] and [573 transitivity].\n\nA reflexive relation that is also transitive is called a [Preorder preorder].',
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