If not P implies Q and not P implies R, what can be deduced about Q and R?

• A. Q implies R
• B. Q and R are both false
• C. No definite relation; Q and R may be independent
• D. R implies Q

Answer: (C)

The key idea is that both statements say: if P is false, then Q is true and R is true. That makes Q and R each follow from not P, so not P is a sufficient condition for both. But there’s no information about Q and R when P is true, so their truth values can vary independently in that case. Because there’s no universal rule linking Q to R across all possibilities, we can’t deduce a relation like Q implies R or R implies Q. The only definite point is that when P is false, both Q and R hold, and otherwise nothing specific about their relationship. Hence there’s no definite relation between Q and R.