I was just wondering what happens with the transitivity in the following examples, sorry if its stupid I'm a newbie student :) .
$R = \{ (0,1) , (2,3) \} $ it is transitive right ? because there is $aRb$ but not $bRc$ so we can't find counter-example so its transitive by default.
and in this example : $R = \{ (1,1) , (2,2) \}$ its symmetric , anti-symmetric and transitive right (because we can relate to $(1,1)$ as $aRb$ and $(1,1)$ as $aRc$ and then $(1,1)$ is $aRc$).
am I assuming wrongly or its correct? Thank you and have a nice day.
Essentially correct. A useful mechanical way to look at vacuous truth of implications is to use the identity $[A\to B]\equiv [B\lor \neg A]$, so that, for instance, $$\forall a,\forall b,\forall c,((aRb\land bRc)\to aRc)$$
is the same as $$\forall a,\forall b,\forall c,(aRc\lor\neg (aRb)\lor \neg (bRc))$$