AMC12 2003 A

AMC12 2003 A · Q6

AMC12 2003 A · Q6. It mainly tests Absolute value, Symmetry.

Define $x \heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$. Which of the following statements is not true?

定义 $x \heartsuit y$ 为 $|x-y|$，对所有实数 $x$ 和 $y$。以下哪个陈述不正确？

(A) x♥y = y♥x for all x and y x♥y = y♥x for all x and y

(B) 2(x♥y) = (2x)♥(2y) for all x and y 2(x♥y) = (2x)♥(2y) for all x and y

(D) x♥x = 0 for all x x♥x = 0 for all x

(E) x♥y > 0 if x ≠ y x♥y > 0 if x ≠ y

Answer

Correct choice: (C)

正确答案：（C）

Solution

We start by looking at the answers. Examining statement C, we notice: $x \heartsuit 0 = |x-0| = |x|$ $|x| eq x$ when $x<0$, but statement C says that it does for all $x$. Therefore the statement that is not true is $\boxed{\mathrm{(C)}\ x\heartsuit 0=x\ \text{for all}\ x}$

我们先看选项。检查陈述 C，我们注意到： $x \heartsuit 0 = |x-0| = |x|$ 当 $x<0$ 时，$|x| eq x$，但陈述 C 说对所有 $x$ 都成立。因此不正确的陈述是 $\boxed{\mathrm{(C)}\ x\heartsuit 0=x\ \text{for all}\ x}$

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