On $A = \left \{1, 2, 3, 4 \right \}$
$\left \{(1,1), (2,1), (1,2)\right \}$ is NOT reflexive because there's no $(2,2)$ in the set. It is symmetric. However, it is NOT transitive.
I'm confused as to why it is not transitive. I thought since $1 R 2$ and $2 R 1$ and $1 R 1$, then the set is transitive?
In another example:
$\left \{ (1,1), (2,2), (3,3), (4,4), (1,4) \right \}$ is reflexive, NOT symmetric because $(4,1)$ is not in the set. However, it is transitive. $1 R 4$ and $4 R 4$ and $1 R 4$. Why is this example transitive but not the former?
It’s not transitive because $\color{brown}2\,R\,1$ and $1\,R\,\color{green}2$, but $\color{brown}2\,\not R\,\color{green}2$. There is no similar failure of transitivity for the second relation.