Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 3 / b / Solution

An upper bound of $S$ is an element $u\in A$ with $s⪯u$ for every $s\in S$. It need not belong to $S$. A maximum, or greatest element, is an $m\in S$ with that same property. A least upper bound is an upper bound $\ell \in A$ satisfying $\ell ⪯u$ for every upper bound $u$.

A maximum belongs to $S$; a least upper bound need not. In a partially ordered set, an existing maximum is automatically the least upper bound, and the antisymmetric relation condition makes each unique.