Please see the below equation.
As the title suggests I have been trying to work out if the following Binary Relation is Transitive.
I have been able to conclude that the relation is Anti-symmetric and Reflexive due to having loops at the end of each vertex and not having a Symmetric nature.
Let $X = \{1, 2, 3, 4\}$
Consider binary relation $R = \{(1,1),(2,1),(2, 2),(3, 1),(3, 3),(4, 1), (4, 2), (4, 4)\}$
On
The relationship is transitive. With a case this small it is easily enough to trace through all the examples and check that $ (a,b) $ and $ (b,c) $ implies $(a,c)$.
Take each pair of pairs if they match up check for the completion of it.
On
The key to this is to look for a counter example: a bridge without end in the relation $$\text{If }\exists a\,\exists b\,\exists c\,[~(a,b){\in}R \wedge (b,c){\in}R \wedge (a,c){\notin}R~]\text{ then the relation is not transitive}$$
Of course this means you have to check exhaustively to ensure you have not missed any.
I can't find any.
Remember the definition: $R$ is transitive iff for any $a, b, c$, if $(a, b)$ and $(b, c)$ are in $R$, then so is $(a, c)$. If $R$ is not transitive, you should be able to find a counterexample.
Of course, this makes showing that $R$ is transitive a bit messy: you have to convince yourself that no counterexample exists. For general $R$ this can be quite complicated, but for this $R$ it's not too bad. Make a list of all the triples $a, b, c$ such that $(a, b)$ and $(b, c)$ are each in $R$; now check each of these.