Consider a sample space $\Omega$ and a sigma-algebra $F$ over it. Let $A$ be a class of $F$-sets. $A$ then generates a partition of $\Omega$ based on the following equivalence relation: Points $w_{1}$ and $w_{2}$ of $\Omega$ are related if, for every set $B$ in $A$, they both belong or they both don't belong to $B$ [ref: Billingsley, 3rd edition, Chapter 1, Section 4, Subfields]. Further, this partition is the same as that generated by the sigma-algebra of $A$ [$\sigma(A)$] defined using the same relation.
Q1. Does it mean that $\sigma(A)$ gives exactly the same information that $A$ gives? In other words, is knowing either $A$ or its sigma-algebra enough to extract any information that we could possibly seek?
Q2. Is there any condition for the above to hold? For example, if $A$ is not a pi-system (not closed under finite intersections) does it hold?
Q3. Can independence of 2 classes $C$ and $D$ (or the lack of it) be inferred by studying their sigma-algebras? For example, if a set in $D$ lies in $\sigma(C)$, is it enough to ensure that $C$ and $D$ are not independent? If so, why?