# Transitivity of a relation (unique)

Okay experts, i do know what the definition of a transitive relation is and how to judge if a relation is transitive or not.But i got across this one question which shouldnt be transitive but the answer in my book says it is .Can someone please tell me how would that be true?

Consider the set A = {1, 2, 3} and the relation R = {(1, 2), (1, 3)}. R is a transitive relation.

• It looks vacuously transitive. To fail to be transitive, you would have to find pairs $(a,b),(b,c)\in R$ with $(a,c)\notin R$, and no such pairs exist. – lulu Mar 3 '17 at 17:13
• is that much enough? i was thinking the same but did not have the faith in my logic.Thanks ! – Zlatan Mar 3 '17 at 17:15
• No problem. Statements that are vacuously true tend to be unintuitive (and sometimes unsatisfying). "Every prime number of the form $2^{3n}+1$ with $n>0$ is a counterexample to Fermat's last Theorem" is a true statement because there are no such primes. – lulu Mar 3 '17 at 17:22

Note that the definition of transitivity says the following: for each $(a,b), (b,c)$ in the relation $R$, we have that $(a,c)$ is an element of the relation. However, in your case there are no elements $(a,b), (b,c)$, so the relation satisfies the condition to be transitive.
Note that if we would have that $R = \{(1,2), (3,1)\}$, then the relation would not be transitive, since we have that $(3,1), (1,2) \in R$, but $(3,2)$ is not in the relation.
I remember transitivity in the following way: I picture the elements $(a,b)$ of a relation is follows: there is a (directed) arrow from $a$ to $b$. So transitivity means that if there is an arrow from $a$ to $b$ and an arrow from $b$ to $c$, then there is an arrow from $a$ to $c$. The moment you have an arrow from $a$ to $b$ and from $b$ to $c$, but no arrow from $a$ to $c$, the relation is not transitive.