This is a continuation of a previous question .
Define that elements of $x=(x_1,x_2,...,x_n)$ are distinct if for each $i≠j, x_i≠x_j.$ Consider a system of linear equation $Ax=b$. We want to understand the property of $(A,b)$ such that the solution $x$ given $A$ and $b$ is distinct in the sense defined above.
In a previous question, Theo kindly showed that the set of such $(A,b)$ is open and dense. Now we want to extend the result in a probabilistic (or measure-theoretic) sense. For example,
Suppose we pick A and b randomly. (For example, suppose that elements of A and b are chosen according to independent uniform distribution on [0,1].) Then calculate x when possible. Now, can it be that x has distinct elements in the sense defined in (a) with probability 1?
I also want to understand whether Sard's theorem can be applied to this setting. I will appreciate your comments.