Symmetric difference

The symmetric difference between two sets A and B (denoted AΔB) is given as: $A\Delta B=(A\backslash B)\cup (B\backslash A)$. This operator is symmetric in that AΔB=BΔA. Also, AΔA is the empty set, while the symmetric difference of the empty set with any set is that set.

The set-theoretic operator Δ obeys several laws:

$A\Delta (C\backslash B)=(A\Delta C)\backslash B$

$A\Delta \cup _{i\in I}B_{i}=\cup _{i\in I}A\Delta B_{i}$

$A\Delta \cap _{i\in I}B_{i}=\cap _{i\in I}A\Delta B_{i}$

$A\Delta (B\Delta C)=(A\Delta B)\cap (A\Delta C)$

Symmetric differences are used to create filters for forcing constructions.