Question

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."

Short answer

1. The logical proposition of the given statement is determined as ${\mathbf{r}}^{\mathbf{\wedge }}\mathbf{~}\mathbf{p}$.
2. The logical proposition of the given statement is determined as ${\mathbf{r}}^{\mathbf{\wedge }}\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}$.
3. The logical proposition of the given statement is determined as ~r→~q.
4. The logical proposition of the given statement is determined as (~p^r)→q.

Step-by-step solution

Introduction to the Concept

"If-then" is a logical sign. It can be depict as $\mathbf{\to }$ .

“whenever__and” is a logical sign. It can be depict as $\mathbf{\wedge }\mathbf{\to }$.

“if__not” is a logical sign. It can be depict as $\mathbf{~}\mathbf{\to }\mathbf{~}$.

"not" is a logical sign. It can be depict as $\mathbf{\bigsqcup }$.

Solution Explanation

a)

The given statements are,

P: The user enters a valid password

q : Access is granted

r: The user has paid the subscription fee

From the given, the statement as follows,

The user has paid the subscription fee, but does not enter a valid password.

Because the terms "but" and "not"are employed in this statement.

The term "but" here refers to "and," therefore it's a conjunction statement, while "not" refers to a negation statement.

In propositional logic, the statement is ${\mathbf{r}}^{\mathbf{\wedge }}\mathbf{~}\mathbf{p}$.

Solution Explanation

b)

From the given, the statement as follows,

Access is granted whenever the user has paid thesubscription fee and enters a valid password.

Because the terms "whenever" and "and"are employed in thisstatement.

The term "whenever" here refers to "if," therefore it's a conjunction statement, while "and" refers to a disjunction statement.

In propositional logic, the statement is ${\mathbf{r}}^{\mathbf{\wedge }}\mathbf{(}\mathbf{p}\mathbf{\to }\mathbf{q}\mathbf{)}$.

Solution Explanation

c)

From the given, the statement as follows,

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

Because the terms "if" and "not"are employed in thisstatement.

Theterm "if" here refers to "if," therefore it's a conditional statement, while "not" refers to a negation statement.

In propositional logic, the statement is ~r→~q .

Solution Explanation

d)

From the given, the statement as follows,

If the user has not entered a valid password but haspaid the subscription fee, then access is granted.

Because the terms "if" and "not", "but" and "then"are employed in thisstatement.

The term "if” and “then”therefore it's a conditional statement, while"but" refers to a conjunction statement.

In propositional logic, the statement is(~p^r)→q.