Let $X$ be a set. We can define the equivalence relation on the power set $2^{X}$ in the following way: the sets $A,B\in 2^{X}$ are equivalent if the symmetric difference $A\triangle B$ is finite.
Questions: Is there any information about this equivalence relation? Is there a standard notation for this equivalence relation?
Let $P_f(X)$ be the set of finite subsets of $X$. As shown in this answer, $(P_f(X), \Delta)$ is a commutative group which acts on the right on $P(X)$ by symmetric difference: \begin{align} P(X) \times P_f(X) & \to P(X) \\ (E,F) &\mapsto E \mathrel{\Delta} F \end{align} Let $\sim$ be the equivalence relation on $P(X)$ defined by $A \sim B$ if and only if $A \mathrel{\Delta} B$ is finite. Then the $\sim$-class of a subset $A$ of $X$ is the orbit of $A$ under this action, since $B = A \mathrel{\Delta} (A \mathrel{\Delta} B)$.
I agree with Daniel Wainfleet that $=_\text{fin}$ is a reasonable notation for $\sim$.