I've been trying to relate a set-valued metric scpace with an inner product. However I'm not sure wheter my attempt is correct or I need to consider something else. Initially I've adapted my solution from this answer Euclidean distance and dot product: Given the distance $\langle x-y,x-y\rangle = \langle x,x\rangle + \langle y,y\rangle - 2\langle x,y\rangle$ (defined on $\mathbb{R}^n$), and that I've defined the symmetric difference-based distance as $\mu(X,Y)=|X\triangle Y|=|X\cup Y\setminus X\cap Y|$, with $X,Y\in(\Omega,\Lambda,\mu)$ being sets, then I only made analogies to translate the distance on $\mathbb{R}^n$ into the distance on sets assumed to be elements of some $(\Omega,\Lambda,\mu)$, where $\Omega$ is a set of elementals and $\Lambda$ is a $\sigma-$algebra.
That is, if $|X|=\langle X,X\rangle$ and $|Y|=\langle Y,Y\rangle$, then $$ |X\triangle Y| = |X| + |Y| - 2\langle X,Y\rangle, $$ so, is it possible to show that $$ \langle X,Y\rangle=\frac{1}{2}(|X|+|Y|-|X\triangle Y|);$$ ?
Your formulation looks fine as long as all sets are finite. As a simplification, note your $\langle X, Y\rangle = |X\cap Y|$
If you've had some basic measure theory, it may help to generalize it to the inner product space $L^2(\Omega, \mu)$ where you are using $\mu = \text{counting measure}$ on the subset of indicator functions.