Tutorial Contents Tutorial Eight: Relations- Reflexivity - Symmetry -Transitivity - Connectedness -
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- Eqivalent Relations
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A relation, Rxy, (that is, the relation expressed by "Rxy") is reflexive in a domain just if there is no dot in its graph without a loop i.e. just if everything in the domain bears the relation to itself. Expressed formally,
Rxy is reflexive just if "xRxx


Examples: "x=y" is reflexive in every domain; "x is the same age as y" is reflexive in the domain of living things, but not, however, in a domain which contains numbers since, presumably, the number 5 is not the same age as itself, since it isn't any age at all.

So notice that a relation may have a property in one domain which it lacks in others.


Rxy is irreflexive in a domain just if no dot in its graph has a loop.

That is:


Rxy is irreflexive just if "xRxx

Examples: "x is older than y"; "x>y".


Rxy is non-reflexive just if it is neither reflexive nor irreflexive i.e. at least one of the dots in its graph has a loop and at least one does not. That is:
Rxy is non-reflexive just if [$xRxx$xRxx].


(We could have said: just if ["xRxx"xRxx]. Remember that ""xj" is equivalent to "$xj".)


Examples: "x likes y" (in a suitable domain); "x=y2" (in a domain containing the natural numbers, for instance).


Notice that in the empty domain a relation will be both reflexive and irreflexive: since there will be no dots at all, it follows that there will be no dots with loops (so it is irreflexive) and no dots without loops (so it will be reflexive).

Remember also that in the empty domain anything beginning """ will be true (as long as it contains no names, in which case the formula can't have an interpretation with an empty domain).



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