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| A 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." |
A 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." |
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| For example, the relation $<$ (``less than") is an irreflexive relation on the set of natural numbers. |
For example, the relation $<$ (``less than") is an irreflexive relation on the set of natural numbers. |
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| Note that ``irreflexive" is not simply the negation of ``\PMlinkname{reflexive}{Reflexive} |
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} |
." 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. |
and irreflexive, it is easy to come up with relations that are neither. |
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| 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 $K$. If $G$ is any group having a subgroup which is not self-normalizing (for example, if $G$ is a nontrivial abelian group), then there is a (proper) subgroup $H$ of $G$ such that $\neg H\mathcal{R} H$. So this relation $\mathcal{R}$ is non-reflexive. Moreover, since the normalizer of any group is the group itself, we have $G\mathcal{R} G$. So $\mathcal{R}$ is non-irreflexive. |
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