Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Discrete mathematics, continuous mathematics, and probability / 3 / c

An element $u\in A$ is a least upper bound of $S$ if it is an upper bound, meaning $s\le u$ for every $s\in S$, and $u\le v$ for every upper bound $v\in A$ of $S$. The candidate need not belong to $S$. Because a partial order is an antisymmetric relation, two least upper bounds must be equal. A least upper bound is also called the supremum.

An element $m$ is a maximum of $S$ if $m\in S$ and $s\le m$ for every $s\in S$. Thus it is an upper bound that is itself a member. Every maximum is the least upper bound: any upper bound must lie above $m$ because $m\in S$. A supremum outside $S$ need not be a maximum.