When thinking about transitive relations, I came up with this "theorem":
All relations $\mathrm{R}: S\to S$ defined on $S = \left\{1, 2\right\}$ are transitive.
But, I couldn't come up with a proof. So, can somebody help me and either prove the above or provide a counter-example for the same?
No. Consider $R=\{(1,2), \, (2,1)\}$.
Then we have $1\,R\,2\,R\,1$ but not $1\,R\,1$.