Definition 1. 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 $Y$,
\[X \symdif Y = \empt \enspace\lrimp\enspace X = Y.\]
Proposition 2. Commutativity. For any sets $X$ and $Y$,
\[X \symdif Y = Y \symdif X.\]Proposition 3. Associativity. For any sets $X$, $Y$ and $Z$,
\[(X \symdif Y) \symdif Z = X \symdif (Y \symdif Z).\]To summarize Proposition 1 (I), commutativity and associativity of $\symdif$:
Proposition 4. For any set $X$, $(\powerset(X), \symdif)$ is an abelian group with neutral element $\empt$.
Proposition 5. For any sets $X$ and $Y$,
\[X \symdif Y = (X \cup Y) \setdif (X \cap Y).\]References
- [1]
- Wikipedia contributors. Symmetric difference. Wikipedia, The Free Encyclopedia.
- [2]
- Thomas Jech. Set Theory. 3rd Millennium ed, rev. and expanded. Springer, 2002. p. 9.