Past exam of the year 1 of the computer science course of the University of Oxford / 2021 / Discrete mathematics, continuous mathematics, and probability / 2 / c / i / Solution

Take $u=(1,1)$ and $v=(1,2)$. Their first coordinates are equal, so both $uRv$ and $vRu$ hold, although $u\ne v$. Thus $R$ is not an antisymmetric relation, and hence it is not a partial order. The failure arises because its comparisons ignore the second coordinate entirely.