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Difference between revisions of "2000 AMC 10 Problems/Problem 21"

(New page: ==Problem== ==Solution== ==See Also== {{AMC10 box|year=2000|num-b=20|num-a=22}})
 
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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?
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  I. All alligators are creepy crawlers.
 +
  II. Some ferocious creatures are creepy crawlers.
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III. Some alligators are not creepy crawlers.
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 +
<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==
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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 <math>A</math>, <math>C</math>, and <math>F</math>.
 +
 +
We got the following information:
 +
* If <math>x</math> is an <math>A</math>, then <math>x</math> is an <math>F</math>.
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* There is some <math>x</math> that is a <math>C</math> and at the same time an <math>A</math>.
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We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are <math>A</math>s, but only Johnny is a <math>C</math>" meets both conditions, but the first statement is false.
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 +
We CAN conclude that the second statement is true. We know that there is some <math>x</math> that is a <math>C</math> and at the same time an <math>A</math>. Pick one such <math>x</math> and call it Bobby. Additionally, we know that if <math>x</math> is an <math>A</math>, then <math>x</math> is an <math>F</math>. Bobby is an <math>A</math>, therefore Bobby is an <math>F</math>. And this is enough to prove the second statement -- Bobby is an <math>F</math> that is also a <math>C</math>.
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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.
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Therefore the answer is <math>\boxed{\text{II only}}</math>.
  
 
==See Also==
 
==See Also==
  
 
{{AMC10 box|year=2000|num-b=20|num-a=22}}
 
{{AMC10 box|year=2000|num-b=20|num-a=22}}

Revision as of 09:51, 11 January 2009

Problem

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

  I. All alligators are creepy crawlers.
 II. Some ferocious creatures are creepy crawlers.
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 $\boxed{\text{II only}}$.

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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