Definition of independence What is the mathematical definition of the independence sign $\perp$?
Example of IV exclusion as (conditional) independence.
\begin{align}
(Y_1,Y_0)\perp Z|X
\end{align}

The definition of "independence" denoted by $\perp$ is not any different than usual. The Wiki on independence is quite extensive if you want more summary information. The gist is that the joint cumulative probability distribution function is equal to the product of the marginal cumulative probability distribution functions.
The expression $(Y_1, Y_0) \perp Z$ means that the CDF of the pair of random variables can be factored from the CDF of $Z$:
$$F_{Y_1, Y_0, Z} = F_{Y_1, Y_0} F_Z$$
which holds to the gist where $\vec U := (Y_1, Y_0)$.
In this case we saying that $(Y_1. Y_0)$ are conditionally independent with $Z$ depending on $X$. So at a given level of $X$ it holds that $(Y_1. Y_0)$ are independent from $Z$, but perhaps not across various levels of $X$.
