Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Digital systems, linear algebra, introduction to formal proof resit / 8 / b

On the universe $\mathbb{Z}$, interpret the first-order signature byThis assigns both constants, a total binary function, a unary relation and a binary relation, so it specifies a complete first-order structure. If $R_2(x)$ and $R_2(y)$ both hold, then $x,y\ge 0$, and hence $f(c_1,c_2)=0\le x+y=f(x,y)$, proving the consequent. If either premise fails, the implication is true. Thus the universally quantified implication holds for every pair of integers, nonvacuously for every pair of nonnegative integers. Distinct constant symbols need not denote distinct objects unless a separate axiom requires it.