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My task is the following:
elements in set $A$: $\{a,b,c,d,e,f\}$
relations between them: $\{(a,b),(b,f),(c,b),(c,d),(d,e),(e,a),(f,d),(f,e)\}$

Question is, is the relation between them antisymmetric and irreflexive?

I am asking because, for antisymmetric I think, the counter-example is: $a \rightarrow b \rightarrow f$ and $f\rightarrow e \rightarrow a$.

And the irreflexive counter-example would be: $a \rightarrow b \rightarrow f \rightarrow e \rightarrow a$.

Or I must look only direct connections, not transitive with elements in between two elements?

Thank you for your time and help.

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  • $\begingroup$ reflexive means xRx for all x (no middles). And antisymmetric means there are not different x,y for which both xRy and yRx (again no middles used). $\endgroup$
    – coffeemath
    May 16, 2016 at 7:49

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The relation $R$ is in fact antisymmetric. Antisymmetry says that whenever $(x,y) \in R$ and $(y,x) \in R$, we must have $x = y$. However, there is no pair $(x,y) \in R$ such that $(y,x) \in R$, so it is vacuously antisymmetric.

It is also irreflexive, because no element is related to itself, meaning that there is no element $x$ such that $(x,x) \in R$.

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