Question

Translate these system specifications into English where the predicate is S(x,y) is “x is in state y” and where the domain for and consists of all systems and all possible states, respectively.

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\exists \mathbf{xS}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{open}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\left(}\mathbf{S}\mathbf{\right(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{malfunctioning}\mathbf{)}\mathbf{}\mathbf{\vee }\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{diagnostic}\mathbf{\left)} \\ \mathbf{c}\mathbf{\right)}\mathbf{}\exists \mathbf{xS}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{open}\mathbf{)}\mathbf{}\mathbf{\vee }\mathbf{\exists }\mathbf{xS}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{diagnostic}\mathbf{)} \\ \mathbf{d}\mathbf{)}\mathbf{}\exists \mathbf{x}\mathbf{\neg }\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{available}\mathbf{\left)} \\ \mathbf{e}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\neg }\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{working}\mathbf{)} \end{gathered}$$

Short answer

a) There exists a system that is in the state open.

b) All systems are in the state malfunction or in the state diagnostic.

c) There exists a system that is in the state open or there exists a system that is in the state diagnostic.

d) There exists a system that is not in the state available.

e) All systems are not in the state working.

Step-by-step solution

Given

S(x,y) is “x is in state y”

Existential qualification: there exist an element x in the domain such that P(x)

Now universal quantification $\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{:}\mathbf{P}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ for all values of x in the domain

Find the condition

S(x,y) = is “x is in state y”

Existential qualification: there exist an element x in the domain such that P(x)

Now, universal quantification$\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$: for all values of n in the domain P(x)

Find the condition

S(x,y) is “x is in state y ”

Existential qualification: there exist an element x in the domain such that P(x)

Now universal quantification$\mathbf{\forall }\mathbf{xp}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$: for all values of x in the domain P(x)

Find the condition

S(x,y) is “x is in state y”

Existential quantification:there exist an element x in the domain such that P(x)

Now universal quantification $\mathbf{\forall }\mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$: for all values of x in the domain P(x)

Find the condition

S(x,y) is “x is in state y”

Existential qualification:there exist an element xin the domain such that P(x)

Now universal quantification$\mathbf{\forall }\mathbf{zP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$: for all values of x in the domain P(x)