Question

如维恩图所示，集合 $A$ 和 $B$ 的元素个数相同。它们的并集有 $2007$ 个元素，交集有 $1001$ 个元素。求集合 $A$ 的元素个数。

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 $A.$

$503$

$1006$

$1504$

$1507$

$1510$

答案：C

知识点：韦恩图容斥原理

难度评级：1290

解答：

设 $A$ 和 $B$ 各有 $x$ 个元素，交集有 $y$ 个元素。

条件给出 $2x - y = 2007$ 且 $y = 1001.$ 将 $y$ 代入第一式得到 $\begin{align*} 2x - 1001 &= 2007 \\ 2x &= 3008 \\ x &= 1504. \end{align*}$

所以正确答案是 C。

Let $x$ be the number of elements in each $A$ and $B.$ Also let $y$ be the number of elements in their intersection.

The conditions give us that $2x - y = 2007$ and $y = 1001.$ Plugging $y$ into the first equation yields $\begin{align*} 2x - 1001 &= 2007 \\ 2x &= 3008 \\ x &= 1504. \end{align*}$

Thus, C is the correct answer.

2007 AMC 8 第 13 题

答案：C

难度评级：1290

解答：

其他年份的第 13 题