2011 AMC 10B Problem 8

The rule is "(at least $80^\circ$ AND sunny) $\Rightarrow$ crowded." An implication is equivalent to its contrapositive: "not crowded $\Rightarrow$ NOT (at least $80^\circ$ AND sunny)."

Negating an "and" statement gives an "or" statement: at least one of the two conditions failed. "Not at least $80^\circ$" means cooler than $80^\circ$. So on June 10, the temperature was cooler than $80^\circ$ F or it was not sunny (possibly both).

Choice (A) is too strong: a cool sunny day is consistent with an empty beach. Choices (C), (D), (E) are conditionals not implied by the rule.

The answer is $\boxed{\textbf{(B)}\ \text{cooler than } 80^{\circ}\text{ F or not sunny}}$.

Two rules of logic settle the problem: the contrapositive of "$P \Rightarrow Q$" is "not $Q \Rightarrow$ not $P$", and De Morgan's law says "not ($P$ and $R$)" is "(not $P$) or (not $R$)". Whenever a hypothesis has several parts, the failure of the conclusion only tells you that some part failed, not which.

Test a concrete scenario: $70^\circ$ and sunny, beach empty. This is allowed by the rule, and it violates (A) and (D), so those are out. (C) and (E) say nothing forced by the rule. Only (B) survives.

• Negating "and" as "and", choosing (A), which claims both conditions failed when only one needs to.
• Choosing (E), which reads like a contrapositive but has the wrong hypothesis (cooler rather than hotter).