The no reflective relations one set containing n element is
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A binary relation is called irreflexive, or anti-reflexive, if it doesn't relate any element to itself. An example is the "greater than" relation (x > y) on the real numbers. 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. However, a relation is irreflexive if, and only if, its complement is reflexive.
A relation ~ on a set X is called quasi-reflexive if every element that is related to some element is also related to itself, formally: ∀ x, y ∈ X : 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. It does make sense to distinguish left and right quasi-reflexivity, defined by ∀ x, y ∈ X : x ~ y ⇒ x ~ x[3] and ∀ x, y ∈ X : x ~ y ⇒ y ~ y, respectively. For example, a left Euclidean relation is always left, but not necessarily right, quasi-reflexive. A relation R is quasi-reflexive if, and only if, its symmetric closure R∪RT is left (or right) quasi-reflexive.
A relation ~ on a set X is called coreflexive if for all x and y in X it holds that if x ~ y then x = y.[4] An example of a coreflexive relation is the relation on integers in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation. The union of a coreflexive relation and a transitive relation on the same set is always transitive. A relation R is coreflexive if, and only if, its symmetric closure is anti-symmetric.
A reflexive relation on a nonempty set X can neither be irreflexive, nor asymmetric, nor antitransitive.
The reflexive closure ≃ of a binary relation ~ on a set X is the smallest reflexive relation on X that is a superset of ~. Equivalently, it is the union of ~ and the identity relation on X, formally: (≃) = (~) ∪ (=). For example, the reflexive closure of (<) is (≤).
The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set X 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 X with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to ~ except for where x~x is true. For example, the reflexive reduction of (≤) is (<).