Question

Express each of these system specifications using predicates, quantifiers, and logical connectives.

a) Every user has access to an electronic mailbox.

b) The system mailbox can be accessed by everyone in the group if the file system is locked.

c) The firewall is in a diagnostic state only if the proxy server is in a diagnostic state.

d) At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode.

Short answer

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\left[}\mathbf{R}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{Q}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{(}\mathbf{T}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\to }\mathbf{S}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{\exists }\mathbf{x}\mathbf{\left(}\mathbf{\right(}\mathbf{V}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{}\mathbf{\wedge }\mathbf{\neg }\mathbf{W}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{\exists }\mathbf{yU}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)} \end{gathered}$$

Step-by-step solution

∀xP(x)

Let P(x) be "user xhas access to electronic mailbox".

Existential quantification$\exists \mathbf{xP}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$: There exists an element x in the domain such that P(x)

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

∀x[R(x)→Q(x)]

Let Q(x) be "The system mailbox can be accessed by users in group x",R(x) be "The file system in group x is locked".

∀x(T(x)→S(x))

Let S(x) be "Firewall of group x is in a diagnostic state" and T(x) be "Proxy server of group x is in a diagnostic state"

∃x((V(x) ∧¬W(x))→∃yU(x,y)

Let U(x,y) be "Router y in group x functions normally" and V(x)be "Throughput of group x is between 10 kbps and 500 kbps" and W(x) be "proxy server of groupx is in diagnostic mode".