Translate these system specifications into English where the predicate S(x, y) is
“x is in state y” and where the domain for x and y consists of all systems and all
possible states, respectively.
a) ∃xS(x, open)
b) ∀x(S(x, malfunctioning) ∨ S(x, diagnostic))
c) ∃xS(x, open) ∨ ∃xS(x, diagnostic)
d) ∃x¬S(x, available)
e) ∀x¬S(x, working)
Given,
"S(x,y)="\\text{x is in state y}""
Negation "\\lnot p:\\ not\\ p"
Disjunction "p\\lor q: p\\ or\\ q"
Existential quantification "\\exists xP(x):" There exists an element x in domain such that P(x)
Universal quantification "\\forall\\ xP(x):P(x)" for all values of x in the domain.
Using the above interpretations, we can translate the prepositions into English.
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.
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