= Solution
Represent a subset by the coordinate <vector> of its <indicator function> in $\mathbb F_2^n$. <Symmetric difference> then becomes coordinatewise addition, and the proposed pairing is the <dot product>
$$
B(x,y)=\sum_{i=1}^nx_iy_i.
$$
It is a <symmetric bilinear form>, and it is <nondegenerate> because its <Gram matrix> in the singleton basis is the identity. Coordinate <permutations> preserve the sum, so the natural $S_n$ action preserves the <bilinear form>.
Let $j=(1,\ldots,1)$, the <vector> representing the full set. Then
$$
W=\langle j\rangle^\perp=\{x:\sum_ix_i=0\}
$$
is the even-weight subspace. For even $n$, $B(j,j)=n=0$ in $\mathbb F_2$, so $j\in W$. Moreover $B(x,x)=\sum_ix_i^2=\sum_ix_i=0$ for $x\in W$, making the restricted <bilinear form> alternating. Nondegeneracy on the ambient space implies $W^\perp=\langle j\rangle$, so the radical of $B|_W$ is exactly
$$
W\cap W^\perp=\langle j\rangle.
$$
Thus it descends to an <alternating bilinear form> on $W/\langle j\rangle$, independent of the representatives, and that quotient form has zero radical. \b[The induced form is <nondegenerate>], and its dimension is $n-2$. This is the <symplectic quotient of the binary subset module>; at $n=2$ it is the zero-dimensional <nondegenerate> space.
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