Use rule of inference to show that [p-(q or r)][(p^~q)-r]?

Question

Use rule of inference to show that [p->(q or r)][(p^~q)->r]?

Answer

Given:

1. p → (q ∨ r)

2. (p ∧ ¬q) → r

To prove:

(p ∧ ¬q) → (q ∨ r)

Proof:

Step 1: Assume (p ∧ ¬q).

Step 2: Using Simplification, we can infer p from (p ∧ ¬q).

Step 3: Using Simplification, we can infer ¬q from (p ∧ ¬q).

Step 4: Using Modus Ponens, we can infer (q ∨ r) from p and p → (q ∨ r).

Step 5: Using Disjunctive Syllogism, we can infer (q ∨ r) from ¬q and (q ∨ r).

Step 6: Using Implication, we can conclude (p ∧ ¬q) → (q ∨ r) from (p ∧ ¬q) and (q ∨ r).

Explanation:

To prove that (p ∧ ¬q) → (q ∨ r) using the given premises, we use the rule of inference to derive the conclusion.

Step 1: Assume (p ∧ ¬q).

We begin the proof by assuming that (p ∧ ¬q) is true.

Step 2: Using Simplification, we can infer p from (p ∧ ¬q).

Since (p ∧ ¬q) is true, we can extract p from it using the Simplification rule.

Step 3: Using Simplification, we can infer ¬q from (p ∧ ¬q).

Similarly, we can extract ¬q from (p ∧ ¬q) using the Simplification rule.

Step 4: Using Modus Ponens, we can infer (q ∨ r) from p and p → (q ∨ r).

From the given premise p → (q ∨ r) and the derived fact that p is true, we can use the Modus Ponens rule to conclude (q ∨ r).

Step 5: Using Disjunctive Syllogism, we can infer (q ∨ r) from ¬q and (q ∨ r).

Since we have ¬q as a derived fact and (q ∨ r) as a given fact, we can apply the Disjunct