Applications of Propositional Logic

Translate the given statement into propositional logic using the propositions

provided.

$1.$ You can graduate only if you have completed the requirements of your

major and you do not owe money to the university and you do not have an

overdue library book. Express your answer in terms of

g: $$You can graduate,$$

m:$$You owe money to the university,$$

r:$$You have completed the requirements of your major,$$ and

b:$$You have an overdue library book.$$

$2.$ Express these system specifications using the propositions

p $$The message is scanned for viruses$$ and

q $$The message was sent from an unknown system$$ together with logical

connectives $($including negations$).$

a$)$The message is scanned for viruses whenever the message was sent

from an unknown system.$$

b$)$The message was sent from an unknown system but it was not scanned

for viruses.$$

c$)$It is necessary to scan the message for viruses whenever it was sent

from an unknown system.$$

d$)$When a message is not sent from an unknown system it is not scanned

for viruses.$$

$3.$ Express these system specifications using the propositions

p $$The user enters a valid password,$$

q $$Access is granted,$$ and

r $$The user has paid the subscription fee$$ and logical connectives $($including

negations$).$

a$)$The user has paid the subscription fee, but does not enter a valid

password.$$

b$)$Access is granted whenever the user has paid the subscription fee and

enters a valid password.$$

c$)$Access is denied if the user has not paid the subscription fee.$$

d$)$If the user has not entered a valid password but has paid the

subscription fee, then access is granted.$$

$4.$ When three professors are seated in a restaurant, the hostess asks them:

$$Does everyone want coffee?$$ The first professor says: $$I do not know.$$

The second professor then says: $$I do not know.$$ Finally, the third

professor says: $$No$,$ not everyone wants coffee. $$The hostess comes back

and gives coffee to the professors who want it$.$ How did she figure out who

wanted coffee?

$5.$ Steve would like to determine the relative salaries of three coworkers using

two facts. First, he knows that if Fred is not the highest paid of the three,

then Janice is$.$ Second, he knows that if Janice is not the lowest paid, then

Maggie is paid the most. Is it possible to determine the relative salaries of

Fred, Maggie, and Janice from what Steve knows? If so$,$ who is paid the

most and who the least? Explain your reasoning.