Reflexive Relation

The reflexivity of a binary relation R on a set is given when x R x for all elements x of the quantity applies (ie, each element is in relation to itself ). R is called reflexive. The relation is called irreflexive if the relation x R x for any element x of the quantity (ie, no element is in relation to itself ).

The nature of the contrast between reflexive and irreflexive is contrary but not contradictory, because there are relations which are neither reflexive nor irreflexive.

The reflexivity is one of the prerequisites for an equivalence relation or an order relation; the irreflexivity is one of the prerequisites for a strict order relation.

Formal definition

Is a set and a binary relation on, then we define (using infix notation ):

Examples

Reflexive

  • The less-than - equal relation on the real numbers is reflexive, since it is always true. She is also a total order. The same applies to the relation.
  • The usual equality on the real numbers is reflexive, since it is always true. She is also an equivalence relation.
  • The subset relation between sets is reflexive, since it is always true. She is also a partial order.

Irreflexive

  • The less-than relation on the real numbers is irreflexive, since never applies. She is also a strict total order. The same applies to the relation.
  • The inequality on the real numbers is irreflexive, since never applies.
  • The proper subset relation between sets is irreflexive, since never applies. She is also a strict partial order.

Neither reflexive nor irreflexive

  • The following relation on the set of real numbers is neither reflexive nor irreflexive: ( Reason: To apply for apply. )

Representation as a directed graph

Any relation on a set can be regarded as a directed graph (see example above). The nodes of the graph are the elements of. From node to node is a directed edge if and only pulled (an arrow ) if the following applies.

The reflexivity of the graph can now be characterized as follows: For each node, there is a loop. Accordingly, the irreflexivity is given by the fact that there is a loop for a node.

Properties

  • By means of the same relation ( which consists of all pairs ) can also be characterized as the terms:   is reflexive   is irreflexive
  • Is the relative reflective and irreflexive, this also applies to the converse relation. EXAMPLES The converse relationship is to be, to be converse.
  • If the relation is reflexive, then the complementary relation is irreflexive. Is irreflexive, then is reflexive. The complementary relation is defined by.
  • The relation on the empty set is the only relation both reflexive irreflexive.
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