Past exam of the year 1 of the computer science course of the University of Oxford / 2005 / Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation / 8 / a / Solution

In this first-order structure, $R(x,z)\wedge R(y,z)$ says $x\le z$ and $y\le z$. Thus $z$ is an upper bound of $x,y$. The universally quantified clause says that if $w$ is any other upper bound, then $z\le w$. Therefore $z$ is their least upper bound. In the usual total order on the natural numbers, this is their maximum:This describes a condition on the values assigned to the free variables $x,y,z$; $\varphi$ itself is not a closed first-order sentence.