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AMC10每日一题（2000年真题#21）

2000 AMC 10 竞赛试题/第21题

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 $\boxed{\text{(B) \, II only}}$ .