Question

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

(a)At least one console must be accessible during every fault condition.

(b)The e-mail address of every user can be retrieved whenever the archive contains at least one message sent by every user on the system.

(c)For every security breach there is at least one mechanism that can detect that breach if and only if there is a process that has not been compromised.

(d)There are at least two paths connecting every two distinct endpoints on the network.

(e)No one knows the password of every user on the system except for the system administrator, who knows all passwords.

Short answer

$$\begin{gathered} \left(\mathrm{a}\right)\forall \mathrm{y}\exists \mathrm{xA}(\mathrm{x},\mathrm{y}) \\ \left(\mathrm{b}\right)\forall \mathrm{x}\exists \mathrm{y}\left(\mathrm{B}\right(\mathrm{x},\mathrm{y})\to \mathrm{C}(\mathrm{x}\left)\right) \\ \left(\mathrm{c}\right)\forall \mathrm{y}\exists \mathrm{xD}(\mathrm{x},\mathrm{y})\leftrightarrow \exists \mathrm{z}\exists \left(\mathrm{z}\right) \\ \left(\mathrm{d}\right)\forall \mathrm{y}\forall \mathrm{z}\exists \mathrm{x}\exists \mathrm{a}\left[\mathrm{x}\ne \mathrm{a}\to \left(\mathrm{F}\right(\mathrm{x},\mathrm{y},\mathrm{z})\wedge \mathrm{F}(\mathrm{a},\mathrm{y},\mathrm{z}\left)\right)\right] \\ \left(\mathrm{e}\right)\forall \mathrm{x}\left[\mathrm{H}\left(\mathrm{x}\right)\to \forall \mathrm{yG}(\mathrm{x},\mathrm{y})\right]\wedge -\exists \mathrm{x}\left[\neg \mathrm{H}\left(\mathrm{x}\right)\wedge \forall \mathrm{yG}(\mathrm{x},\mathrm{y})\right] \end{gathered}$$

Step-by-step solution

Definition of quantifier

$\forall Quantifiersaretermsthatcorrespondtoquantitiessuchas"some"or"all"andindicatethenumberofitemsforwhichacertainpropositionistrue.Forexpres\sin gthegivenstatementsintermsofpredicates,quantifiersand\log icalconnectives,unders\tan d$Quantifiersare terms that correspond to quantities such as "some" or "all" and indicate the number of items for which a certain proposition is true.

For expressing the given statements in terms of predicates, quantifiers and logical connectives, understand the meaning of statements and use proper quantifier. Here, the quantifier “$\forall$” indicates “All” whereas the quantifier “role="math" localid="1668588696048" $\exists$” represents “Some” or “There exists.”

(a) Express the given statement in terms of predicates, quantifiers and logical connectives

At least one console must be accessible during every fault condition.

Supposeandare the domains of all consoles and all fault conditions respectively.

Let A(x,y) be “console x is accessible during fault condition y.”

The symbolic representation of above statement is$\forall \mathrm{y}\exists \mathrm{xA}(\mathrm{x},\mathrm{y})$ .

(b) Express the given statement in terms of predicates, quantifiers and logical connectives

The e-mail address of every user can be retrieved whenever the archive contains at least one message sent by every user on the system.

Supposeandare the domains of all email addresses of users and messages respectively.

Let B(x,y) be “email address of userhas sent messageon the system that is contained in the archive.”

Let C(x) be “email address of usercan be retrieved.”

The symbolic representation of above statement is $\forall \mathrm{x}\exists \mathrm{y}\left(\mathrm{B}\right(\mathrm{x},\mathrm{y})\to \mathrm{C}(\mathrm{x}\left)\right)$.

(c) Express the given statement in terms of predicates, quantifiers and logical connectives

For every security breach there is at least one mechanism that can detect that breach if and only if there is a process that has not been compromised.

Suppose x is the domain of all mechanisms andis the domain of all securitybreaches.

Also, z is the domain of all processes.

Let D(x,y) be “mechanismcan detect security breach.”

Let E(z) be “process z has been compromised.”

The symbolic representation of above statement is $\forall \mathrm{y}\exists \mathrm{xD}(\mathrm{x},\mathrm{y})\leftrightarrow \exists \mathrm{z}\exists \left(\mathrm{z}\right)$.

(d) Express the given statement in terms of predicates, quantifiers and logical connectives

There are at least two paths connecting every two distinct endpoints on the network.

Suppose x is the domain of all paths y andis the domain of all distinct endpoints.

Also, z is the domain of all distinctendpoints.

Let F(x,y,z) be “pathconnects distinct endpoints y andz.”

The symbolic representation of above statement is $\forall \mathrm{y}\forall \mathrm{z}\exists \mathrm{x}\exists \mathrm{a}\left[\mathrm{x}\ne \mathrm{a}\to \left(\mathrm{F}\right(\mathrm{x},\mathrm{y},\mathrm{z})\wedge \mathrm{F}(\mathrm{a},\mathrm{y},\mathrm{z}\left)\right)\right]$.

(e) Express the given statement in terms of predicates, quantifiers and logical connectives

No one knows the password of every user on the system except for the system administrator, who knows all passwords.

Suppose x is the domain of all people and y is the domain of all users.

Also, z is the domain of all distinct endpoints.

Let G(x,y) be “x knows password of user y.”

Let H(x) be “x is a system administrator.”

The symbolic representation of above statement is $\forall \mathrm{x}\left[\mathrm{H}\left(\mathrm{x}\right)\to \forall \mathrm{yG}(\mathrm{x},\mathrm{y})\right]\wedge \neg \exists \mathrm{x}\left[\neg \mathrm{H}\left(\mathrm{x}\right)\wedge \forall \mathrm{yG}(\mathrm{x},\mathrm{y})\right]$.