DEF-SYD. Symmetric Difference.
The symmetric difference between $X$ and $Y$ is
\[X\symd Y = (X\smallsetminus Y)\cup(Y\smallsetminus X).\]
For any $X$,
\[X\symd\varnothing = X.\]For any $X$ and $Y$,
\[X\symd Y = \varnothing \enspace\lrimp\enspace X = Y.\]
For any $X$ and $Y$,
\[X\symd Y=Y\symd X.\]
For any $X$, $Y$ and $Z$,
\[(X\symd Y)\symd Z=X\symd(Y\symd Z).\]
As a result:
For any $X$, $(\powerset(X),\symd)$ is an abelian group with neutral element $\varnothing$.
For any $X$ and $Y$,
\[X\symd Y = (X\cup Y)\smallsetminus (X\cap Y).\]