A binary relationMathworldPlanetmath on a set A is said to be irreflexiveMathworldPlanetmath (or antireflexive) if aA, ¬aa. In other words, “no element is -related to itself.”

For example, the relationMathworldPlanetmath < (“less than”) is an irreflexive relation on the set of natural numbers.

Note that “irreflexive” is not simply the negationMathworldPlanetmath of “reflexiveMathworldPlanetmathPlanetmath ( .” Although it is impossible for a relation (on a nonempty set) to be both reflexive ( and irreflexive, there exist relations that are neither. For example, the relation {(a,a)} on the two element set {a,b} is neither reflexive nor irreflexive.

Here is an example of a non-reflexive, non-irreflexive relation “in nature.” A subgroupMathworldPlanetmathPlanetmath in a group is said to be self-normalizing if it is equal to its own normalizer. For a group G, define a relation on the set of all subgroups of G by declaring HK if and only if H is the normalizer of K. Notice that every nontrivial group has a subgroup that is not self-normalizing; namely, the trivial subgroup {e} consisting of only the identityPlanetmathPlanetmathPlanetmathPlanetmath. Thus, in any nontrivial group G, there is a subgroup H of G such that ¬HH. So the relation is non-reflexive. Moreover, since the normalizer of a group G in G is G itself, we have GG. So is non-irreflexive.

Title irreflexive
Canonical name Irreflexive
Date of creation 2013-03-22 15:41:45
Last modified on 2013-03-22 15:41:45
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 14
Author CWoo (3771)
Entry type Definition
Classification msc 03E20
Synonym antireflexive
Related topic Reflexive