Express these system specifications using the propositions...

Express these system specifications using the propositions $p^{\text {"The user enters a valid password," }} q$ "Access is granted," and $r$ "The user has paid the subscription fee" and logical connectives. 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 subseription fee, then access is granted."

Express these system specifications using the propositions $p^{\text {"The user enters a valid password," }} q$ "Access is granted," and $r$ "The user has paid the subscription fee" and logical connectives.
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 subseription fee, then access is granted."

Key Concepts

Propositional Variables

These are the basic units of propositional logic that represent simple, indivisible statements. In this context, variables like p, q, and r are used to denote distinct system conditions or properties, such as whether a user enters a valid password, whether access is granted, and whether a subscription fee has been paid.

Logical Connectives

Logical connectives are symbols or words used to connect propositions to form more complex expressions. They include operators such as AND, OR, NOT, and IF-THEN. These connectives allow us to combine basic statements in order to capture the relationships and constraints expressed in system specifications.

Conjunction (AND)

The conjunction operator, typically represented by ?, is used to denote that two or more propositions are simultaneously true. In the context of system specifications, it is used when multiple conditions must be met together, such as requiring both a valid password and a paid subscription for access.

Negation (NOT)

Negation is a logical operator used to reverse the truth value of a proposition. When a statement is negated, it expresses that the proposition is false. This operator is essential in system specifications to denote conditions like 'does not enter' or 'has not paid'.

Implication (IF-THEN)

The implication operator, usually represented by ?, is used to create conditional statements. It expresses that if one proposition holds (the antecedent), then another proposition must also hold (the consequent). This mirrors many real-world rules found in system specifications, where certain actions or outcomes depend on particular conditions being met.

Please give Ace some feedback