411. Determine the rational values for $x$ and $y$ in the equation: $\lambda^{y}=y^{x}$.
411. Let $y=a x$, i.e., $(a x)^{x}=x^{a x}$ or $a x=x^{a} ; a=x^{a-1}$, from which $x=a^{\frac{1}{a-1}}$, then $y=a^{\frac{a}{a-1}} ;$ if we set $\frac{1}{a-1}=n$, we get $a=\frac{n+1}{n}$, consequently.

$$
x=\left(\frac{n+1}{n}\right)^{n} \text { and } y=\left(\frac{n+1}{n}\right)^{n+1}
$$

12 According to the problem. Collection of problems.

![](https://cdn.mathpix.com/cropped/2024_05_21_926f0bb262c8f9569516g-179.jpg?height=603&width=557&top_left_y=247&top_left_x=384)

Thus, $\subseteq 0$.