Question

Let p, q and r be statements. Suppose you know that the statement form ((q → p) ∨ r) ∨ (∼ r ∧ p) is false. What can you conclude about the truth values of the three statement variables?

Answer 1

Let $|((q → p) ∨ r) ∨ (∼ r ∧ p)|=F.$ By the definition of disjunction, we have that $|(q → p) ∨ r|=F$ and $|∼ r ∧ p|=F.$ It follows from the definition of disjunction that $|q → p|=F$ and $|r|=F.$ By definition of implication we get that $|q|=T$ and $|p|=F.$ In this case, $|∼ r ∧ p|=T\land F=F.$

We conclude that the statements $p$ and $r$ are false and the statement q is true.