Question

Translate these specifications into English where F(p) is “Printer is out of service,” B(p) is “Printer p is busy,” L(j) is “Print job j is lost,” and Q(j) is “Print job j is queued.”

$$\begin{gathered} \mathbf{a}\mathbf{)}\mathbf{}\mathbf{\exists }\mathbf{p}\mathbf{(}\mathbf{F}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\wedge }\mathbf{B}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{\exists }\mathbf{jL}\mathbf{(}\mathbf{j}\mathbf{\left)} \\ \mathbf{b}\mathbf{\right)}\mathbf{}\mathbf{\forall }\mathbf{pB}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\to }\mathbf{\exists }\mathbf{jQ}\mathbf{\left(}\mathbf{j}\mathbf{\right)} \\ \mathbf{c}\mathbf{)}\mathbf{}\mathbf{\exists }\mathbf{j}\mathbf{}\mathbf{(}\mathbf{Q}\mathbf{\left(}\mathbf{j}\mathbf{\right)}\mathbf{\wedge }\mathbf{L}\mathbf{(}\mathbf{j}\mathbf{}\mathbf{)}\mathbf{)}\mathbf{\to }\mathbf{\exists }\mathbf{pF}\mathbf{}\mathbf{(}\mathbf{p}\mathbf{\left)}\mathbf{\right)} \\ \mathbf{d}\mathbf{)}\mathbf{}\mathbf{(}\mathbf{\forall }\mathbf{pB}\mathbf{\left(}\mathbf{p}\mathbf{\right)}\mathbf{\wedge }\mathbf{\forall }\mathbf{jQ}\mathbf{\left(}\mathbf{j}\mathbf{\right)}\mathbf{)}\mathbf{\to }\mathbf{\exists }\mathbf{jL}\mathbf{(}\mathbf{j}\mathbf{)} \end{gathered}$$

Short answer

a) If there is a printer that is out of service and is busy, then there is a print job that is lost.

b) If all printers are busy, then there is a print job that is queued.

c) If there exists a print job that is lost and queued, then there exists a printer that is out of service.

d) If all printers are busy and all print jobs are queued, then there exists a print job that is lost.

Step-by-step solution

Find the condition

a) If there is a printer that is out of service and is busy,

$\to$Means IF-THEN, $\wedge$means AND and$\exists$means THERE then there is a print job that is lost. EXISTS .

Find the condition

b) If all printers are busy, then there is a print job that is

→ Means IF-THEN, $\forall$means ALL and $\exists$means THERE queued. EXISTS.

Find the condition

c) If there exists a print job that is lost and queued,

$\to$Means IF-THEN, $\wedge$means AND and $\exists$means THERE then there exists a printer that is out of service. EXISTS.

Find the condition

d) If all printers are busy and all print jobs are queued,

$\to$Means IF-THEN, role="math" localid="1668576871200" $\wedge$means AND, $\exists$ means THERE then there exists a print job that is lost. EXISTS and means ALL.