Question

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

(a)Every user has access to exactly one mailbox.

(b)There is a process that continues to run during all error conditions only if the kernel is working correctly.

(c)All users on the campus network can access all websites whose url has a .edu extension.

(d)There are exactly two systems that monitor every remote server.

Short answer

$$\begin{gathered} \left(\mathrm{a}\right)\forall \mathrm{x}\exists \mathrm{y}\left(\mathrm{A}\right(\mathrm{x},\mathrm{y})\wedge \forall \mathrm{z}(\mathrm{A}(\mathrm{x},\mathrm{z})\to \mathrm{y}=\mathrm{z}\left)\right) \\ \left(\mathrm{b}\right)\exists \mathrm{x}\left(\mathrm{B}\right(\mathrm{x})\to \forall \mathrm{yC}(\mathrm{x},\mathrm{y}\left)\right) \\ \left(\mathrm{c}\right)\forall \mathrm{x}\forall \mathrm{y}\left(\mathrm{E}\right(\mathrm{y},.\mathrm{edu})\to \mathrm{D}(\mathrm{x},\mathrm{y}\left)\right) \\ \left(\mathrm{d}\right)\exists \mathrm{x}\exists \mathrm{z}\left[\mathrm{x}\ne \mathrm{z}\wedge \forall \mathrm{yF}(\mathrm{x},\mathrm{y})\wedge \forall \mathrm{yF}(\mathrm{z},\mathrm{y})\wedge \forall \mathrm{w}(\forall \mathrm{yF}(\mathrm{w},\mathrm{y})\to (\mathrm{w}=\mathrm{x}\vee \mathrm{w}=\mathrm{z}\left)\right)\right] \end{gathered}$$

Step-by-step solution

Definition of quantifier

Quantifiers are 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 properquantifier. Here, the quantifier “$\forall$” indicates “All” whereas the quantifier “role="math" localid="1668595371429" $\exists$” represents “Some” or “There exists.”

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

Every user has access to exactly one mailbox.

Supposeandare the domains of all users and mailboxes respectively.

Let A(x,y) be “student x has access to mailbox .y”

The given sentence can be rewritten as “All users have access to a mailbox and the user does not have access to any other mailbox.”

The symbolic representation of above statement is$\forall \mathrm{x}\exists \mathrm{y}\left(\mathrm{A}\right(\mathrm{x},\mathrm{y})\wedge \forall \mathrm{z}(\mathrm{A}(\mathrm{x},\mathrm{z})\to \mathrm{y}=\mathrm{z}\left)\right)$

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

There is a process that continues to run during all error conditions only if the kernel is working correctly.

Supposeandare the domains of all processes and error conditions respectively.

Let B(x) be “kernel of processis working correctly.”

Let C(x,y) be “processruns during error condition.”

The given sentence can be rewritten as “There exists a process, such that if the kernel of the process is working correctly, then the process continues to run during all error conditions.”

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

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

All users on the campus network can access all websites whose url has a .edu extension.

Suppose x is the domain of all users on the campus network.

Also, y is the domain of all websites and z is the domain of all url extensions.

Let D(x,y) be “usercan access website y.”

Let D(y,z) be “website has url extension z.”

The symbolic representation of above statement is $\forall \mathrm{x}\forall \mathrm{y}\left(\mathrm{E}\right(\mathrm{y},.\mathrm{edu})\to \mathrm{D}(\mathrm{x},\mathrm{y}\left)\right)$.

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

There are exactly two systems that monitor every remote server.

Supposeis the domain of all systems andis the domain of all remote servers.

Let F(x,y) be “systemmonitors remote server y.”

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