Difference between revisions of "2000 AMC 10 Problems/Problem 21"
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==Problem== | ==Problem== | ||
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If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? | If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? | ||
| − | + | <cmath>\textrm{I. All alligators are creepy crawlers.}</cmath> | |
| − | + | <cmath>\textrm{II. Some ferocious creatures are creepy crawlers.}</cmath> | |
| − | + | <cmath>\textrm{III. Some alligators are not creepy crawlers.}</cmath> | |
<math>\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}</math> | <math>\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}</math> | ||
| − | ==Solution== | + | == 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. | 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. | ||
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We CAN NOT conclude that the third statement is true. For example, consider the situation when <math>A</math>, <math>C</math> and <math>F</math> are equivalent (represent the same set of objects). In such case both conditions are satisfied, but the third statement is false. | We CAN NOT conclude that the third statement is true. For example, consider the situation when <math>A</math>, <math>C</math> and <math>F</math> 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 <math>\boxed{\text{II only}}</math>. | + | Therefore the answer is <math>\boxed{\text{(B) \, II only}}</math>. |
==See Also== | ==See Also== | ||
Latest revision as of 03:13, 1 February 2024
Problem
If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?
![\[\textrm{I. All alligators are creepy crawlers.}\]](http://latex.artofproblemsolving.com/7/2/b/72b807a06e111f8fdb87f07064fff3c3cfc5fee3.png)
![\[\textrm{II. Some ferocious creatures are creepy crawlers.}\]](http://latex.artofproblemsolving.com/c/7/3/c73e126dd37b7f7dd433101545605fcbc09b70ab.png)
![\[\textrm{III. Some alligators are not creepy crawlers.}\]](http://latex.artofproblemsolving.com/f/4/0/f40481a58c634ac2f60a79caafb1ed5aebeb3240.png)
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 
, 
, and 
.
We got the following information:
- If
is an 
, then is an 
. - There is some that is a
and at the same time an .
We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are s, but only Johnny is a " meets both conditions, but the first statement is false.
We CAN conclude that the second statement is true. We know that there is some 
that is a and at the same time an . Pick one such and call it Bobby. Additionally, we know that if is an , then is an . Bobby is an , therefore Bobby is an . And this is enough to prove the second statement -- Bobby is an that is also a .
We CAN NOT conclude that the third statement is true. For example, consider the situation when , and 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 
.
See Also
| 2000 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 20 |
Followed by Problem 22 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
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