Let's break down the problem step-by-step to determine the truth value of the expression \(\sim[q \vee(\sim s \wedge q)]\), where \(q\) and \(s\) both represent true statements.
1. Identify Statements:
- \(q\) is true.
- \(s\) is true.
2. Negation:
- \(\sim s\) represents the negation of \(s\). Since \(s\) is true, \(\sim s\) is false.
3. Logical AND (Conjunction):
- \(\sim s \wedge q\) represents the conjunction of \(\sim s\) and \(q\). Since \(\sim s\) is false and \(q\) is true, the result of the conjunction is false because an AND operation is true only if both operands are true.
4. Logical OR (Disjunction):
- \(q \vee (\sim s \wedge q)\) represents the disjunction of \(q\) and \(\sim s \wedge q\). Since \(q\) is true and \(\sim s \wedge q\) is false, the result of the disjunction is true because an OR operation is true if at least one of the operands is true.
5. Negation of the Whole Expression:
- \(\sim [q \vee (\sim s \wedge q)]\) represents the negation of the entire expression \(q \vee (\sim s \wedge q)\). Since \(q \vee (\sim s \wedge q)\) is true, negating it results in false.
Thus, the truth value of the expression \(\sim[q \vee (\sim s \wedge q)]\) is false.
Now, let's consider the statement “Is the statement \(\sim [q \vee (\sim s \wedge q)]\) true or false?”
From our step-by-step analysis above, we determined that the statement \(\sim [q \vee (\sim s \wedge q)]\) is false.
Therefore:
False is the truth value of the statement.
