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Version 5 |
| A binary relation $\mathcal{R}$ on a set $A$ is said to be \emph{irreflexive} 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} 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, it is easy to come up with relations that are neither. |
and irreflexive, it is easy to come up with relations that are neither. |
