Is the relation $R = \{a,b\}$ transitive? I was studying relations and functions and transitive relations were defined as :
$$R=\{(x,y)\} ; xRy ~\text{and}~ yRz \implies xRz$$
But what about a single element $(a,b)$? 
For example, if the relation was
$$R = \{(1,2),(2,1),(1,1),(2,2)\}$$
Then clearly, it is transitive, but if the relation were :
$$R = \{(1,2),(2,1),(1,1),(2,2),(5,6)\}$$
Or simply :
$$R=\{(5,6)\}$$
Then are these relations transitive ?

Yup, that's transitive! For every triple of elements $a, b, c$, if $aRb$ and $bRc$ then $aRc$. It doesn't matter that there are some elements - namely, $5$ and $6$ - which can't be "extended" to form a triple. There's no more to transitivity than the definition you wrote above.
