show that the following propositional logics are equivalent using propositional equivalence laws. State the laws that you are applying.
¬[q∧ ( p → ¬ q ) ] → p ≡ ( p ⋁ q )
To show that the propositional logic ¬[q∧ ( p → ¬ q ) ] → p is equivalent to ( p ⋁ q ) using propositional equivalence laws, we will perform a step-by-step analysis. Let's break it down:
Step 1: Expand the implication ( p → ¬ q ) to its logical equivalent expression ¬p ⋁ ¬q.
- This step utilizes the equivalence law known as Material Implication.
The expression ¬[q∧ ( p → ¬ q ) ] → p becomes ¬[q ∧ ( ¬p ⋁ ¬q )] → p.
Step 2: Use De Morgan's laws to distribute ¬ over the conjunction (q ∧ ( ¬p ⋁ ¬q )).
- De Morgan's laws state that ¬(p ⋁ q) ≡ ¬p ∧ ¬q and ¬(p ∧ q) ≡ ¬p ⋁ ¬q.
The expression ¬[q ∧ ( ¬p ⋁ ¬q )] becomes (¬q ∨ (p ∧ q)).
Step 3: Apply the implication equivalence law ( ¬p ∨ q ) → r ≡ ( p ⋁ q ) ∨ r.
- This is one of the implications equivalence laws.
The expression (¬q ∨ (p ∧ q)) → p becomes (¬q ∨ (p ∧ q)) ∨ p.
Step 4: Apply the associative law to group the disjunctions.
- The associative law states that p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r.
The expression (¬q ∨ (p ∧ q)) ∨ p becomes ((¬q ∨ p) ∨ q) ∨ p.
Step 5: Apply the commutative law to rearrange the disjunctions.
- The commutative law states that p ∨ q ≡ q ∨ p.
The expression ((¬q ∨ p) ∨ q) ∨ p becomes ((p ∨ ¬q) ∨ q) ∨ p.
Step 6: Apply the tautology law p ∨ ¬p ≡ T (always true).
- The tautology law means that any proposition p or its negation ¬p is always true.
The expression ((p ∨ ¬q) ∨ q) ∨ p becomes (T ∨ q) ∨ p.
Step 7: Apply the tautology law T ∨ q ≡ T (always true).
The expression (T ∨ q) ∨ p becomes T ∨ p.
Step 8: Apply the tautology law T ∨ p ≡ T (always true).
The expression T ∨ p is equivalent to T (always true).
Hence, we have shown that ¬[q∧ ( p → ¬ q ) ] → p is equivalent to ( p ⋁ q ) using the propositional equivalence laws: Material Implication, De Morgan's laws, Implication Equivalence, Associative Law, Commutative Law, and Tautology Law.