Lets say I have the following set: $$ \{1, 2\}$$ and on it the following relation is given: $$\{(1, 2), (2, 1)\}.$$
Now is the above relation transitive?
My confusion: we can see, that it is possible to go from $1$ to $2$ and $2$ to $1$. So is it OK to assume that it is possible to go from $1$ to $1$ without having an explicit relation of $(1,1)$?
The relation is neither transitive nor reflexive. It is only symmetric.
Remember that the condition for transitivity is that if $a \sim b$ and $b \sim c$, then $a \sim c$.
We have $1 \sim 2$ and $2 \sim 1$. If transitivity held, we would expect $1 \sim 1$ and $2 \sim 2$. But our relation does not contain $(1,1)$ or $(2,2)$ so it is not transitive.
The relation is also obviously not reflexive, because it does not contain $(1,1)$ and $(2,2)$.
Moreover, note that transitivity does not necessarily imply reflexivity: consider $\{(a,b),(b,c),(a,c)\}$ which is transitive, but neither reflexive nor symmetric. (However, a relation that is both transitive and symmetric will be reflexive: I'll leave proving this fact to you ask an exercise.)
