Example 2 If $x \in\left\{-1,1,2, x^{2}-2 x\right\}$, then the set of real numbers $x$ is
Solve: Since $x \in \left\{-1,1,2, x^{2}-2 x\right\}$, we have $x=-1$ or $x=1$ or $x=2$ or $x=$ $x^{2}-2 x$.

When $x=x^{2}-2 x$, $x=0$ or $x=3$, but when $x=1$, $x^{2}-2 x=-1$, which contradicts the distinctness of the elements in the set. Therefore, the set of real numbers $x$ is $\{-1,2,0,3\}$.
Key Points for Solving: Elements in a set are distinct; verify this when solving.