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 / d / Solution

The first two clauses now require $x=2z$ and $y=2z$. Thus they already imply $x=y$. Choose, for example, $x=1,y=2$; no natural number $z$ can satisfy both equations. Consequently the existential quantifier fails for this pair, irrespective of the universally quantified final clause, andIn fact even $x=y=1$ fails, since $1=2z$ has no solution among the natural numbers.