# 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 ?

• You might want to change the question in the title to the one actually being asked. (First and foremost, replace "set" by "relation".) – martin.koeberl May 10 '17 at 1:34

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.
• Then what is '$c$' here ( in my example)? Is it defined to be transitive or there is any explanation? – Jaideep Khare May 10 '17 at 1:23
• @JaideepKhare You're misunderstanding the definition. The definition just says whenever you have such a triple $a, b, c$, then [stuff]. It doesn't say that every element is part of some triple. The only way a relation $S$ is non-transitive is if there are three elements $a, b, c$ with $aSb$, $bSc$, but not $aSc$ - can you find such a triple of elements in this case? – Noah Schweber May 10 '17 at 1:24