State whether each of the following propositions is a tautology or a contradiction or contingent (i.e. neither).You must give a (brief) reason to justify each of your answers.(a) P(b) (P ∧ Q) → (P → Q)(c) ¬(P → Q) → (¬Q)(d) (¬Q ∧ (P → Q)) → ¬P(e) ((P → Q) ∧ (Q → R)) ↔ (P → R)(f) (R ∨ P ) → (P ∨ (Q ∨ R))

(a) P: This is contingent. The truth value of P depends on the assignment of P. It can be either true or false.

(b) (P ∧ Q) → (P → Q): This is a tautology. If P and Q are both true, then P → Q is also true. If P is false, then P → Q is true (since a false statement implies anything). If Q is false, then P ∧ Q is false, and a false statement implies anything.

(c) ¬(P → Q) → (¬Q): This is contingent. The truth value depends on the assignment of P and Q. For example, if P is true and Q is false, then ¬(P → Q) is true and ¬Q is also true, making the whole statement true. But if P and Q are both true, then ¬(P → Q) is false and ¬Q is also false, making the whole statement false.

(d) (¬Q ∧ (P → Q)) → ¬P: This is a tautology. If ¬Q is true, then P → Q is false, which makes ¬Q ∧ (P → Q) false, and a false statement implies anything. If ¬Q is false (i.e., Q is true), then P → Q is true, which makes ¬Q ∧ (P → Q) false again, and a false statement implies anything.

(e) ((P → Q) ∧ (Q → R)) ↔ (P → R): This is a tautology. It's the transitivity of implication. If P implies Q and Q implies R, then it must be that P implies R.

(f) (R ∨ P ) → (P ∨ (Q ∨ R)): This is a tautology. The right side of the implication is always true because it includes all elements of the left side. Therefore, regardless of the truth values of P, Q, and R, the whole statement is always true.

(a) P: This is contingent. The truth value of P depends on the specific truth value assigned to P. It can be either true or false.

(b) (P ∧ Q) → (P → Q): This is a tautology. If P and Q are both true, then P → Q is also true. If either P or Q is false, then (P ∧ Q) is false, which makes the whole statement true because a false statement implies anything.

(c) ¬(P → Q) → (¬Q): This is contingent. The truth value depends on the specific truth values assigned to P and Q. For example, if P is true and Q is false, the whole statement is true. But if P and Q are both true, the whole statement is false.

(d) (¬Q ∧ (P → Q)) → ¬P: This is a tautology. If Q is false and P implies Q, then P must be false. If either ¬Q or P → Q is false, then the whole statement is true because a false statement implies anything.

(e) ((P → Q) ∧ (Q → R)) ↔ (P → R): This is a tautology. It's a formalization of the transitive property: if P implies Q and Q implies R, then P implies R.

(f) (R ∨ P ) → (P ∨ (Q ∨ R)): This is a tautology. The statement is saying that if either R or P is true, then either P, Q, or R is true. This is always the case because if either R or P is true, then the right side of the implication is automatically true.