The equivalent statement without parentheses is: (2) p ∨ q .
De Morgan's Laws to Simplify the Statement
De Morgan's Laws state that:
The negation of a conjunction (AND) is equivalent to the disjunction (OR) of the negations: ¬(p ∧ q) ≡ ¬p ∨ ¬q
The negation of a disjunction (OR) is equivalent to the conjunction (AND) of the negations: ¬(p ∨ q) ≡ ¬p ∧ ¬q
We want to use these laws to simplify the statement:
(1) ¬(¬p ∧ ¬q)Step 1: Apply De Morgan's Law for Conjunction:
Since we have a negation of a conjunction, we can use the first law:
¬(¬p ∧ ¬q) ≡ ¬¬p ∨ ¬¬q
Step 2: Apply Double Negation Law:
Double Negation Law says that:
¬(¬p) ≡ p
Applying this law to both terms in the expression:
¬¬p ∨ ¬¬q ≡ p ∨ q
Therefore, the equivalent statement without parentheses is:(2) p ∨ q