Final answer:
The statement (p ∧ q ∧ ¬ r) ∨ (p ∧ ¬ q ∧ ¬ r) = p ∧ ¬ r is true. Applying the distributive law to factor out p and ¬ r, and recognizing that q ∨ ¬ q is a tautology (always true), the expression simplifies to p ∧ ¬ r.
Step-by-step explanation:
To determine if the statement (p ∧ q ∧ ¬ r) ∨ (p ∧ ¬ q ∧ ¬ r) = p ∧ ¬ r is true, we can apply logical identities and simplify the expression. Using the distributive law of logic, we can factor out p and ¬ r. Here's the simplification step-by-step:
- (p ∧ q ∧ ¬ r) ∨ (p ∧ ¬ q ∧ ¬ r)
- p ∧ ¬ r ∧ (q ∨ ¬ q)
- p ∧ ¬ r ∧ (True)
- p ∧ ¬ r
Since q ∨ ¬ q is a tautology and always true, the expression p ∧ ¬ r ∧ true simplifies to p ∧ ¬ r. So the statement is True (option a).