Question

对所有实数 $x$ 和 $y$ ，定义 $x \diamond y$ 为 $|x - y|$ 。求下式的值：

$(1 \diamond (2 \diamond 3)) - ((1 \diamond 2) \diamond 3)\text{？}$

Define $x \diamond y$ to be $|x - y|$ for all real numbers $x$ and $y.$ What is the value of

$(1 \diamond (2 \diamond 3)) - ((1 \diamond 2) \diamond 3)?$

$-2$

$-1$

$0$

$1$

$2$

先计算括号内的 $\diamond$ ，并使用 $a \diamond b = |a-b|$ 。

Evaluate the inner $\diamond$ first, using $a \diamond b = |a-b|$

分别计算 $1 \diamond (2 \diamond 3)$ 和 $(1 \diamond 2) \diamond 3$ ，再相减。

Compute $1 \diamond (2 \diamond 3)$ and $(1 \diamond 2) \diamond 3$ separately, then subtract

因为 $2 \diamond 3 = |2-3| = 1$ ，所以 $1 \diamond (2 \diamond 3) = 1 \diamond 1 = 0$ 。

因为 $1 \diamond 2 = |1-2| = 1$ ，所以 $(1 \diamond 2) \diamond 3 = 1 \diamond 3 = |1-3| = 2$ 。

所求值为 $0 - 2 = -2$ 。

所以正确答案是 A。

Since $2 \diamond 3 = |2-3| = 1,$ we get $1 \diamond (2 \diamond 3) = 1 \diamond 1 = 0.$

Since $1 \diamond 2 = |1-2| = 1,$ we get $(1 \diamond 2) \diamond 3 = 1 \diamond 3 = |1-3| = 2.$

The value is $0 - 2 = -2.$

Thus, the correct answer is A.