DEF-SYMDIF. Symmetric Difference.
The symmetric difference between $X$ and $Y$ is defined as
\[X\mathbin{\triangle}Y = (X\smallsetminus Y)\cup(Y\smallsetminus X).\]
PROP-SYMDIF-EMP.
For any $X$:
- $X\mathbin{\triangle}\varnothing=X$.
- $X\mathbin{\triangle}X=\varnothing$.
PROP-SYMDIF-DIF.
For any $X$ and $Y$,
\[X\mathbin{\triangle}Y = (X\cup Y)\smallsetminus (X\cap Y).\]
PROP-SYMDIF-COM. Commutativity.
For any $X$ and $Y$,
\[X\mathbin{\triangle}Y=Y\mathbin{\triangle}X.\]
PROP-SYMDIF-ASS. Associativity.
For any $X$, $Y$ and $Z$,
\[(X\mathbin{\triangle}Y)\mathbin{\triangle}Z=X\mathbin{\triangle}(Y\mathbin{\triangle}Z).\]