Transitive Relations: Special case

Sure, the binary relation $R = \{(1,2), (3,4)\}$ on the set $A = \{1,2,3,4\}$ is transitive.

Indeed, the transitive property is vacuously true for the relation $R$: since there are no $x,y,z$ in $A$ such that $(x,y) \in R$ and $(y,z) \in R$, there is nothing to check.


Yes, your relation is indeed transitive. The nonexistence of any triples $x,y,z$ such that $(x,y),(y,z)$ are both in your relation does not matter. Being transitive merely says that if such a triple existed then you must also have $(x,z)$ in the relation.

Reworded, a relation is transitive iff for any "directed path" (if they even exist in the first place) you can take from one element to another you can also get there by using a single step instead. Since all of your directed paths are of length $1$ anyways, there is nothing more to check.

Tags:

Discrete Mathematics

Elementary Set Theory

Relations

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