Question

Justify each step with the name of the logical equivalence that was used to prove that (⥛q ^ (p → q)) → ⥛p is a taultology.

Answer 1

To prove that (⥛q ^ (p → q)) → ⥛p is a tautology, logical equivalences such as the implication equivalence, De Morgan's law, the associative property, the idempotent property, the distributive property, the negation law, the identity law, and the commutative property are used to justify each step.

Step-by-step explanation:

To prove that (⥛q ^ (p → q)) → ⥛p is a tautology, we can use logical equivalences. Here are the steps:

1. Start with the given expression: (⥛q ^ (p → q)) → ⥛p
2. Use the implication equivalence: (p ∧ q) → r is equivalent to ¬(p ∧ q) ∨ r
3. Apply the equivalence to the given expression: ¬((⥛q) ∧ (p → q)) ∨ ⥛p
4. Use De Morgan's law: ¬(p ∧ q) is equivalent to ¬p ∨ ¬q
5. Apply De Morgan's law to the expression: (¬⥛q ∨ ¬(p→q)) ∨ ⥛p
6. Use the implication equivalence: ¬(p → q) is equivalent to p ∧ ¬q
7. Apply the equivalence to the expression: (¬⥛q ∨ (p∧¬q)) ∨ ⥛p
8. Use the associative property: (p ∨ q) ∨ r is equivalent to p ∨ (q ∨ r)
9. Apply the associative property: (p ∨ ¬q) ∨ (⥛q ∧ ⥛p)
10. Apply the implication equivalence: (¬q ∨ p) ∨ (¬q ∨ p)
11. Use the idempotent property: p ∨ p is equivalent to p
12. Apply the idempotent property to the expression: (¬q ∨ p)
13. Use the implication equivalence: (p ∧ q) → r is equivalent to ¬(p ∧ q) ∨ r
14. Apply the implication equivalence to the expression: ¬(⥛q ∧ (⥛q ∨ p)) ∨ ⥛p
15. Use De Morgan's law: (¬¬q ∨ ¬(⥛q ∨ p)) ∨ ⥛p
16. Apply De Morgan's law to the expression: (q ∨ ¬(⥛q ∨ p)) ∨ ⥛p
17. Use the implication equivalence: ¬(p ∨ q) is equivalent to ¬p ∧ ¬q
18. Apply the implication equivalence to the expression: (q ∨ (¬⥛q ∧ ¬p)) ∨ ⥛p
19. Use the De Morgan's law again: ¬(p → q) is equivalent to p ∧ ¬q
20. Apply the De Morgan's law again: (q ∨ (¬(p → q) ∧ ¬p)) ∨ ⥛p
21. Use the distributive property: p ∧ (q ∨ r) is equivalent to (p ∧ q) ∨ (p ∧ r)
22. Apply the distributive property to the expression: (q ∨ ((¬p ∨ ¬q) ∧ (¬p ∨ ⥛p)))
23. Use the negation law: ¬p ∨ p is equivalent to true
24. Apply the negation law to the expression: (q ∨ (true ∧ (¬p ∨ ¬q)))
25. Use the identity law: true ∧ q is equivalent to q
26. Apply the identity law to the expression: (q ∨ (¬p ∨ ¬q))
27. Use the commutative property: p ∨ q is equivalent to q ∨ p
28. Apply the commutative property to the expression: (¬p ∨ ¬q) ∨ q
29. Use the negation law: ¬q ∨ q is equivalent to true
30. Apply the negation law to the expression: (¬p ∨ true)
31. Use the identity law: true ∨ p is equivalent to true
32. Apply the identity law to the expression: true

From the steps above, we can see that every step is justified using a logical equivalence such as the implication equivalence, De Morgan's law, the associative property, the idempotent property, the distributive property, the negation law, the identity law, and the commutative property. Therefore, (⥛q ^ (p → q)) → ⥛p is indeed a tautology.