For example, the relation (“less than”) is an irreflexive relation on the set of natural numbers.
Note that “irreflexive” is not simply the negation of “reflexive (http://planetmath.org/Reflexive) .” Although it is impossible for a relation (on a nonempty set) to be both reflexive (http://planetmath.org/Reflexive) and irreflexive, there exist relations that are neither. For example, the relation on the two element set is neither reflexive nor irreflexive.
Here is an example of a non-reflexive, non-irreflexive relation “in nature.” A subgroup in a group is said to be self-normalizing if it is equal to its own normalizer. For a group , define a relation on the set of all subgroups of by declaring if and only if is the normalizer of . Notice that every nontrivial group has a subgroup that is not self-normalizing; namely, the trivial subgroup consisting of only the identity. Thus, in any nontrivial group , there is a subgroup of such that . So the relation is non-reflexive. Moreover, since the normalizer of a group in is itself, we have . So is non-irreflexive.
|Date of creation||2013-03-22 15:41:45|
|Last modified on||2013-03-22 15:41:45|
|Last modified by||CWoo (3771)|