Please explain how this digraph is Reflexive, Symmetric and Transitive

Question

Please explain how this diagram is reflexive, symmetric and transitive.

Answer 1

It appears that the diagram is saying each element is only related to itself, so $R= \{(1,1),(2,2),(3,3),(4,4)\}$. This is clearly a reflexive relation. So let's look at the other two properties.

1. A Relation is symmetric if $(a,b) \in R$ implies $(b,a) \in R$.

2. A Relation is transitive if $(a,b) \in R$ and $(b,c) \in R$ implies $(a,c)\in R$.

Note that (1) is true since each relation looks like $(x,x)$ and in addition to $(x,x)$, the relation $(x,x)$ with the two $x$'s switched is in $R$. Note also that (2) is true because anytime we have two relations whose first and second coordinate overlap, like $(\_,b)$ and $(b,\_)$ they are the relation $(x,x)$, which then only requires $(x,x)$ to be in $R$. It is, so we're good.