Express this system specifications using predicates, quantifiers, and
logical connectives, if necessary.There are exactly two systems that monitor every remote server.
My solution is :
∃x∃y∀z(x != y ∧ (∀s M(z,s) → (z = x ∨ z = y))), where M(a, b) means that system a monitors remote server b
So can anyone please check my solution if it's correct and if not what is the correct solution?
Best Answer
The problem:
Can be rewritten into:
$\color{red}{\text{There is a system x and there is a system z such that the two systems are different}}$ and $\color{blue}{\text{system x monitors all remote servers and system z monitors all remote servers}}$ and $\color{green}{\text{no other system w monitors all remote servers}}$.
$$\color{red}{\exists x,z(x\neq z} \land \color{blue}{\forall y M(x,y) \land \forall y M(z,y)}\land \color{green}{\lnot\exists w(w \neq x \land w \neq z \land \forall y M(w,y))}\color{red}{)}$$
Since $\lnot\exists w(P(w)) \iff \forall w(\lnot P(w))$, we can rewrite this as:
$$\color{red}{\exists x,z(x\neq z} \land \color{blue}{\forall y M(x,y) \land \forall y M(z,y)}\land \color{green}{\forall w(\underbrace{w = x \lor w = z}_{\psi} \lor \lnot\underbrace{\forall y M(w,y)}_{\phi})}\color{red}{)}$$
Finally, using the property:
$\psi \lor \lnot\phi \iff \phi\rightarrow \psi$
we end up with the accepted solution:
$$\color{red}{\exists x,z(x\neq z} \land \color{blue}{\forall y M(x,y) \land \forall y M(z,y)}\land \color{green}{\forall w(\forall y M(w,y) \rightarrow (w = x \lor w = z))}\color{red}{)}$$