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.