"is the reciprocal of"...over the set of non-zero real numbers is: (a) Symmetric (b) Reflexive (c) Transitive (d) Equivalence
I know that it is definitely symmetric, as, in (a, b), if a is the reciprocal of b, then obviously b is the reciprocal of a. Also, it is not reflexive, as no number can be reciprocal of itself, therefore none of the element is related to itself. I think this relation should be transitive, because, for a relation to be transitive, if (a, b) belong to R and (b, c) belong to R, then (a, c) should also belong to R. Also, a, b, and c should be distinct. In this relation, we cannot find two pairs of the format (a, b) and (b, c) such that a, b, and c are different. Therefore, the question of proving that (a, c) does not belong to R does not arise. Hence the relation should be transitive. I'm basing this on what I read in one of the answers on this website only that we should think in this manner, "a relation is not transitive only if we can find something which makes it non transitive".
Please help.
It is not transitive. $2$ is the reciprocal of $\frac 1 2$ and $\frac 1 2$ is the reciprocal of $ 2$ but $2$ is not the reciprocal of $2$.