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 $(1,1)$?

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    $\begingroup$ No the set is not transitive. The relation does not say that 1 is related to 1 (there is no (1,1) in the set). Transitivity does not imply reflexivity. $\endgroup$ – shyamupa Jan 26 '14 at 5:36

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.)


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