Past exam of the year 1 of the computer science course of the University of Oxford / 2003 / Calculus, probability, linear algebra, discrete mathematics, logic and proof, and models of computation / 8 / b

Let $\mathbf{A}$ be the one-element partially ordered set $\{a\}$ with $R(a,a)$. Taking $x=a$ witnesses the existential quantifier, and the only $y$ is comparable with $a$. Thus $\mathbf{A}\models \phi$.

Let $\mathbf{B}$ be the two-element antichain $\{a,b\}$, with $R$ consisting only of $(a,a)$ and $(b,b)$. This is a partial order. If $x=a$, choose $y=b$, and if $x=b$, choose $y=a$. In either case both $R(x,y)$ and $R(y,x)$ are false. Therefore no witness for the existential quantifier exists: