# 2000 AMC 10 Problems/Problem 21

## Problem

If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? $$\textrm{I. All alligators are creepy crawlers.}$$ $$\textrm{II. Some ferocious creatures are creepy crawlers.}$$ $$\textrm{III. Some alligators are not creepy crawlers.}$$ $\mathrm{(A)}\ \text{I only} \qquad\mathrm{(B)}\ \text{II only} \qquad\mathrm{(C)}\ \text{III only} \qquad\mathrm{(D)}\ \text{II and III only} \qquad\mathrm{(E)}\ \text{None must be true}$

## Solution

We interpret the problem statement as a query about three abstract concepts denoted as "alligators", "creepy crawlers" and "ferocious creatures". In answering the question, we may NOT refer to reality -- for example to the fact that alligators do exist.

To make more clear that we are not using anything outside the problem statement, let's rename the three concepts as $A$, $C$, and $F$.

We got the following information:

• If $x$ is an $A$, then $x$ is an $F$.
• There is some $x$ that is a $C$ and at the same time an $A$.

We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are $A$s, but only Johnny is a $C$" meets both conditions, but the first statement is false.

We CAN conclude that the second statement is true. We know that there is some $x$ that is a $C$ and at the same time an $A$. Pick one such $x$ and call it Bobby. Additionally, we know that if $x$ is an $A$, then $x$ is an $F$. Bobby is an $A$, therefore Bobby is an $F$. And this is enough to prove the second statement -- Bobby is an $F$ that is also a $C$.

We CAN NOT conclude that the third statement is true. For example, consider the situation when $A$, $C$ and $F$ are equivalent (represent the same set of objects). In such case both conditions are satisfied, but the third statement is false.

Therefore the answer is $\boxed{\text{(B) \, II only}}$.

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