Question

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

a) At least one mail message, among the nonempty set of messages, can be saved if there is a disk with more than 10 kilobytes of free space.

b) Whenever there is an active alert, all queued messages are transmitted.

c) The diagnostic monitor tracks the status of all systems except the main console.

d) Each participant on the conference call whom the host of the call did not put on a special list was billed.

Short answer

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{\exists }\mathbf{xP}\mathbf{(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{\exists }\mathbf{yQ}\mathbf{(}\mathbf{y}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{\exists }\mathbf{x}\mathbf{\left[}\mathbf{R}\mathbf{\right(}\mathbf{x}\mathbf{)}\mathbf{\to }\mathbf{\forall }\mathbf{y}\mathbf{(}\mathbf{S}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{)}\mathbf{\left)}\mathbf{\right]} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{(}\mathbf{x}\mathbf{\ne }\mathbf{mainconsule}\mathbf{\leftrightarrow }\mathbf{T}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\left)} \\ \mathbf{d}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{x}\mathbf{\left[}\mathbf{\right(}\mathbf{U}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\wedge }\mathbf{V}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{W}\mathbf{(}\mathbf{x}\mathbf{\left)}\mathbf{\right]} \end{gathered}$$

Step-by-step solution

∃xP(x)→∃yQ(y)

Let P(x) be "disk x has more than 10 kilobytes of free space" andQ(y) be "Message y can be saved".

∃x[R(x)→∀y(S(x,y))]

Let R(x) be "Computer x has an active alert", S(x,y) be "Queued message y of computerxis transmitted".

∀x(x≠mainconsule↔T(x))

Let T(x) be "The diagnostic monitor tracks the status of systemx".

∀x[(U(x)∧V(x))→W(x)]

Let U(x) be" x participated on the conference call" andV(x) be" x was not put on the special list" and W(x) be "x was billed"..