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.
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 $(x,x) \in R$.