Answer
a) r ∧ ¬p
b) r ∧ p → q
c) ¬r → ¬q
d) ¬p ∧ r → q
Work Step by Step
The propositions are :
p = " The user enters a valid password"
q = "Access is granted"
r = " The user has paid the subscription fee"
a) " The user has paid the subscription fee, but does not enter a valid password."
The first part of the sentence is exactly the proposition r which states : " The user has paid the subscription fee. "BUT" can also be replaced by " AND". And "does not enter a valid password" is the negation of the proposition p. So we can actually rephrase this as: " The user has paid the subscription fee AND he does not enter a valid password".
b) " Access is granted whenever the user has paid the subscription fee and enter a valid password."
The second part of the sentence, you can read the AND. So, we already know that there's going to be the r ∧ p which is stating that the user has paid the subscription fee and enter a valid password. Now the tricky part is to figure out :
- If access is granted then the user has paid the subscription fee and enter a valid password
OR
- If the user has paid the subscription fee and enters a valid password, then access is granted.
You need to know which one is sufficient and which one is necessary. So, paying the subscription fee and entering a valid password would be a necessary condition. And if you have the access granted it is a sufficient condition to know that you did pay the subscription fee and you have entered a valid password. So the answer is r ∧ p is necessary for q, and q is sufficient for r ∧ p which is stating that " IF the user has paid the subscription fee AND enters a valid password, THEN access is granted. ".
c) "Acces is denied if the user has not paid the subscription fee"
This can simply be rephrased as " IF the user has not paid the subscription fee, THEN access is denied".
**It is important to note that the opposite(converse) is not necessarily true. You cannot write: ¬q→¬r , because this means: IF access is denied, THEN the user has not paid the subscription fee. Pay attention to the fact that it doesn't have the same meaning, because if access is denied, it could be because the user didn't put a valid password, but it doesn't necessarily mean that the user didn't pay the subscription fee. BUT, if the user didn't pay the subscription fee, THEN it is true and always true that access will be denied.
d) " If the user has not entered a valid password but has paid the subscription fee, then access is granted."
This sentence is the exact translation of the propositions into English. IF + negation of p + BUT which can be replaced with AND + r + THEN + q.
Which would be ¬p ∧ r → q.
