AMC8 2007

AMC8 2007 · Q13

AMC8 2007 · Q13. It mainly tests Linear equations, Logic puzzles.

Sets A and B, shown in the Venn diagram, have the same number of elements. Their union has 2007 elements and their intersection has 1001 elements. Find the number of elements in A.

集合 A 和 B 如维恩图所示，具有相同数量的元素。它们的并集有 2007 个元素，交集有 1001 个元素。求集合 A 中的元素个数。

(A) 503 503

(B) 1006 1006

(D) 1507 1507

(E) 1510 1510

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Let $C$ denote the set of elements that are in $A$ but not in $B$. Let $D$ denote the set of elements that are in $B$ but not in $A$. Because sets $A$ and $B$ have the same number of elements, the number of elements in $C$ is the same as the number of elements in $D$. This number is half the number of elements in the union of $A$ and $B$ minus the intersection of $A$ and $B$. That is, the number of elements in each of $C$ and $D$ is \[ \frac{1}{2}(2007-1001)=\frac{1}{2}\cdot 1006=503. \] Adding the number of elements in $A$ and $B$ to the number in $A$ but not in $B$ gives $1001+503=1504$ elements in $A$.

（C）设 $C$ 表示属于 $A$ 但不属于 $B$ 的元素集合。设 $D$ 表示属于 $B$ 但不属于 $A$ 的元素集合。由于集合 $A$ 和 $B$ 的元素个数相同，$C$ 中元素的个数与 $D$ 中元素的个数相同。这个数等于 $A$ 与 $B$ 的并集的元素个数减去交集的元素个数后的一半。也就是说，$C$ 和 $D$ 中每个集合的元素个数为 \[ \frac{1}{2}(2007-1001)=\frac{1}{2}\cdot 1006=503. \] 将 $A$ 与 $B$ 的元素个数加上“在 $A$ 中但不在 $B$ 中”的元素个数，得到 $A$ 中共有 $1001+503=1504$ 个元素。

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