The symmetric difference between two sets $X$ and $Y$ is
\[X \symdif Y = (X \setdif Y) \cup (Y \setdif X).\]
For any set $X$,
\[X \symdif \empt = X.\]For any sets $X$ and $Y$,
\[X \symdif Y = \empt \enspace\lrimp\enspace X = Y.\]
For any sets $X$ and $Y$,
\[X \symdif Y = Y \symdif X.\]
For any sets $X$, $Y$ and $Z$,
\[(X \symdif Y) \symdif Z = X \symdif (Y \symdif Z).\]
As a corollary of Proposition 1 (I), commutativity and associativity of $\symdif$:
For any set $X$, $(\powerset(X), \symdif)$ is an abelian group with neutral element $\empt$.
For any sets $X$ and $Y$,
\[X \symdif Y = (X \cup Y) \setdif (X \cap Y).\]