2000 AMC 10 Problem 21

P21-25~3 minPrint

Logic Puzzles Sets, Estimation & Miscellaneous Test small/specific values or special cases to find or verify the answer

If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?

$\textrm{I. All alligators are creepy crawlers.}$ $\textrm{II. Some ferocious creatures are creepy crawlers.}$ $\textrm{III. Some alligators are not creepy crawlers.}$

Show solution without answering

Key insight

Only statement II must be true: a creepy crawler that is an alligator is also ferocious. Statements I and III each fail in a small made-up example.

Solution

"Must be true" means true in every situation that fits the two facts. To show a statement must be true, reason it out from the facts. To show it need not be true, find one small example where the facts hold but the statement fails.

The two facts:
- Every alligator is ferocious.
- At least one creepy crawler is an alligator.

Statement II. By the second fact, there is some creature, call it Al, that is both a creepy crawler and an alligator. By the first fact, Al is ferocious. So Al is a ferocious creature that is a creepy crawler. Statement II must be true.
Statement I. Imagine a world with two alligators, Al and Bo, both ferocious. Al is a creepy crawler; Bo is not. Both facts hold. But "all alligators are creepy crawlers" is false, because of Bo. So I does not have to be true.
Statement III. Imagine a world with just one alligator, Al, who is ferocious and a creepy crawler. Both facts hold. But "some alligators are not creepy crawlers" is false, since the only alligator is one. So III does not have to be true.

Only II must be true.

The answer is $\boxed{\textbf{(B)}\ \text{II only}}$ .

Why this works

A single example where the facts hold but a statement fails is enough to rule that statement out. In logic problems, "some" means "at least one," and it does not rule out "all." Also ignore what you know about real alligators; only the given facts count.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Abbreviate: every $A$ (alligator) is an $F$ (ferocious); at least one $C$ (creepy crawler) is an $A$ .

Statement II: take a creature that is both $C$ and $A$ . Being an $A$ , it is an $F$ . So there is a creature that is both $F$ and $C$ , and II must be true.

Statement I: build a world with two alligators, only one of which is a creepy crawler. Both hypotheses hold, yet I is false, so I is not forced.

Statement III: build a world with exactly one alligator, and it is a creepy crawler. Both hypotheses hold, yet III is false, so III is not forced.

The answer is $\boxed{\textbf{(B)}\ \text{II only}}$ .

Review in the AoPS books

Intermediate Counting & Probability, Chapter 2: Sets and Logic (§2.4)
Prealgebra, Chapter 7: Ratios, Conversions, and Rates (§7.4)