Question

Give a Boolean expression consisting of only Ps, Qs, ¬s, ∧s, and ∨s which is logically equivalent to the Boolean expression below

¬ (P ↔ Q)

Answer 1

The Boolean expression ¬ (P ↔ Q) is equivalent to (¬P ∨ ¬Q) ∧ (P ∨ Q), using only NOT, AND, and OR operators.

Step-by-step explanation:

We are given the Boolean expression ¬ (P ↔ Q) and we need to find a logically equivalent expression using only P, Q, ¬ (NOT), ∧ (AND), and ∨ (OR) operators. The biconditional (P ↔ Q) is equivalent to (P ∧ Q) ∨ (¬P ∧ ¬Q), meaning P and Q are both true or both false. Therefore, its negation ¬ (P ↔ Q) is equivalent to negating the entire expression, which leads to (¬P ∨ ¬Q) ∧ (P ∨ Q). This result is the disjunction of the negations ANDed with the disjunction of the original statements.