A binary relation on a set is said to be irreflexive (or antireflexive) if , . In other words, “no element is -related to itself."

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 ." 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 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.