Symmetric Difference

SYD#DEF. Symmetric Difference.

The symmetric difference between $X$ and $Y$ is

\[X\symd Y = (X\smallsetminus Y)\cup(Y\smallsetminus X).\]

SYD#PROP-EMP.

1. For any $X$,

   \[X\symd\varnothing = X.\]

2. For any $X$ and $Y$,

   \[X\symd Y = \varnothing \enspace\lrimp\enspace X = Y.\]

SYD#PROP-COM. Commutativity.

For any $X$ and $Y$,

\[X\symd Y = Y\symd X.\]

SYD#PROP-ASS. Associativity.

For any $X$, $Y$ and $Z$,

\[(X\symd Y)\symd Z = X\symd(Y\symd Z).\]

SYD#PROP-GRP.

$(\V,\symd)$ is an abelian group with neutral element $\varnothing$.

Proof.By commutativity, associativity and SYD#PROP-EMP.

SYD#PROP-D.

For any $X$ and $Y$,

\[X\symd Y = (X\cup Y)\smallsetminus (X\cap Y).\]