AMC10 2007 B
AMC10 2007 B · Q5
AMC10 2007 B · Q5. It mainly tests Logic puzzles.
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
在某个国度，所有 Arogs 都是 Brafs，所有 Crups 都是 Brafs，所有 Dramps 都是 Arogs，所有 Crups 都是 Dramps。以下哪个陈述是由这些事实蕴含的？
(A)
All Dramps are Brafs and are Crups.
所有Dramp都是Braf并且是Crup。
(B)
All Brafs are Crups and are Dramps.
所有Braf都是Crup并且是Dramp。
(C)
All Arogs are Crups and are Dramps.
所有Arog都是Crup并且是Dramp。
(D)
All Crups are Arogs and are Brafs.
所有Crup都是Arog并且是Braf。
(E)
All Arogs are Dramps and some Arogs may not be Crups.
所有Arog都是Dramp，有些Arog可能不是Crup。
Answer
Correct choice: (D)
正确答案：（D）
Solution
Answer (D): Let $A$, $B$, $C$, and $D$ represent the following statements about a person in the land.
$A$: Is an Arog. $B$: Is a Braf. $C$: Is a Crup. $D$: Is a Dramp.
Then the statement in the first sentence of the problem can be expressed as:
$A \Rightarrow B,\quad C \Rightarrow B,\quad D \Rightarrow A$ and $C \Rightarrow D.$
The most we can conclude is that
$C \Rightarrow D \Rightarrow A \Rightarrow B.$
So the only statement listed that we are certain is true is that Crups are both Arogs and Brafs.
答案（D）：设 $A$、$B$、$C$、$D$ 表示关于该国某个人的如下陈述。
$A$：是 Arog。 $B$：是 Braf。 $C$：是 Crup。 $D$：是 Dramp。
那么，题目第一句话中的陈述可以表示为：
$A \Rightarrow B,\quad C \Rightarrow B,\quad D \Rightarrow A$ 且 $C \Rightarrow D.$
我们最多能推出
$C \Rightarrow D \Rightarrow A \Rightarrow B.$
因此，在所列陈述中我们唯一能确定为真的是：Crup 同时是 Arog 和 Braf。
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