Question

令 $@$ 表示“与……取平均”的运算： $a @ b = \frac{a+b}{2}$ 。下列哪些分配律对所有数 $x, y$ 和 $z$ 都成立？

I. $x @ (y + z) = (x @ y) + (x @ z)$

II. $x + (y @ z) = (x + y) @ (x + z)$

III. $x @ (y @ z) = (x @ y) @ (x @ z)$

Let $@$ denote the "averaged with" operation: $a @ b = \frac{a+b}{2}.$ Which of the following distributive laws hold for all numbers $x, y,$ and $z?$

I. $x @ (y + z) = (x @ y) + (x @ z)$

II. $x + (y @ z) = (x + y) @ (x + z)$

III. $x @ (y @ z) = (x @ y) @ (x @ z)$

I only

II only

III only

I and III only

II and III only

I 的左边为 $\dfrac{x+y+z}2$ ，右边为 $x+\dfrac{y+z}2$ ，不恒相等。

II 的左边为 $x+\dfrac{y+z}2$ ，右边也为，恒成立。

III 的左边为，右边也为 $\dfrac{2x+y+z}4$ ，恒成立。

所以正确答案是 E。

In statement I, the left-hand side equals $\dfrac{x+y+z}2$ and the right-hand side equals $x+\dfrac{y+z}2,$ so they are not always equal.

In statement II, both sides equal $x+\dfrac{y+z}2,$ so it holds.

In statement III, both sides equal $\dfrac{2x+y+z}4,$ so it also holds.

Thus, the correct answer is E .