Final answer:
The truth value of ~[(p ∧ q) ∧ (p∨~r)] is true when p is true, q is false, and r is true. This is determined by applying logical operators in sequence to the given propositions.
Step-by-step explanation:
If p is true, q is false and r is true, the truth value of ~[(p ∧ q) ∧ (p∨~r)] needs to be calculated. We apply logical operators step-by-step considering the given values:
- p ∧ q is false because q is false and an 'AND' operation (∧) requires both propositions to be true for the result to be true.
- ~r is false because r is true and the negation (~) of a true statement is false.
- p∨~r is true because p is true and an 'OR' operation (∨) requires only one of the propositions to be true for the result to be true.
- Now, we have the 'AND' operation between false and true, which is (false ∧ true) and that equals false.
- Finally, the negation of false is true, so ~[(p ∧ q) ∧ (p∨~r)] is true.
Therefore, the truth value of the expression ~[(p ∧ q) ∧ (p∨~r)] given the values of p, q, and r is true.