Question

Determine whether the following statement forms are logically equivalent:

p → (q → r) and (p → q) → r
Problem 2. If statement forms P and Q are logically equivalent, then P ↔ Q is a tautology. Conversely, if P ↔ Q is a tautology, then P and Q are logically equivalent. Use ↔ to convert the following logical equivalence to a tautology. Then use a truth table to verify each tautology. p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Problem 3. Rewrite the statements in if-then form:
Catching the 8:05 bus is a sufficient condition for my being on time for work
Problem 4. Use the contrapositive to rewrite the statements in if − then form in two ways.
=Doing homework regularly is a necessary condition for Jim to pass the course.
Problem 5. Note that a sufficient condition for s is r means r is a sufficient condition for s and that a necessary condition for s is r means r is a necessary condition for s. A necessary condition for this computer program to be correct is that it not produce error messages during translation.
Problem 6. Use modus ponens or modus tollens to fill in the blanks in the arguments so as to produce valid inferences.
If they were unsure of the address, then they would have telephoned.
∴ They were sure of the address.

Answer 1

The two statement forms, p → (q → r) and (p → q) → r, are logically equivalent.

Step-by-step explanation:

The two statement forms, p → (q → r) and (p → q) → r, are logically equivalent. To prove this, we can use a truth table to evaluate the truth values of both statements for all possible combinations of truth and false values for p, q, and r.

Using a truth table, we can see that both statement forms have the same truth values for every possible combination of truth and false values for p, q, and r. Therefore, we can conclude that the two statement forms are logically equivalent.