irreflexiveIrreflexive2006-02-20 00:58:192008-02-27 18:30:14Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA binary relation $\mathcal{R}$ on a set $A$ is said to be \emph{irreflexive} (or \emph{antireflexive}) if $\forall a\in A$, $\neg a\mathcal{R} a$. In other words, ``no element is $\mathcal{R}$-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 ``\PMlinkname{reflexive}{Reflexive}
." Although it is impossible for a relation (on a nonempty set) to be both \PMlinkname{reflexive}{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 subgroup in a group is said to be \emph{self-normalizing} if it is equal to its own normalizer. For a group $G$, define a relation $\mathcal{R}$ on the set of all subgroups of $G$ by declaring $H\mathcal{R}K$ 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 identity. Thus, in any nontrivial group $G$, there is a subgroup $H$ of $G$ such that $\neg H\mathcal{R} H$. So the relation $\mathcal{R}$ is non-reflexive. Moreover, since the normalizer of a group $G$ in $G$ is $G$ itself, we have $G\mathcal{R} G$. So $\mathcal{R}$ is non-irreflexive.