What Determines the Transitivity of Relations in a Set?

nuuskur · Jan 28, 2015

Having trouble understanding the concept of transitivity.
By definition: If [itex](a,b)\in R\wedge (b,c)\in R \Rightarrow (a,c)\in R[/itex] - Great.

Consider the set [itex]\{a,b\}[/itex]. What makes the relation [itex]\{(a,a)\}[/itex] or [itex]\{(a,a),(a,b)\}[/itex] transitive? How do I translate this in terms of the definition?
What makes an empty set transitive?

mfb · Jan 28, 2015

There is an important part missing in the definition: "For all a,b,c in the set".
You can test all combinations of the set and see if this statement is violated for one combination. If yes, the relation is not transitive. If there is no violation, it is transitive.
For the empty set, there is no combination at all that could violate transitivity, so a relation on it is always transitive.

What Determines the Transitivity of Relations in a Set?

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