Symmetric difference

The symmetric difference between two sets A and B (denoted AΔB) is given as: <math>A\Delta B = (A\backslash B)\cup (B\backslash A)</math>. 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:

<math> A\Delta (C\backslash B) = (A\Delta C)\backslash B </math>

<math> A\Delta \cup_{i\in I} B_i = \cup_{i\in I} A\Delta B_i </math>

<math> A\Delta \cap_{i\in I} B_i = \cap_{i\in I} A\Delta B_i </math>

<math> A\Delta (B\Delta C) = (A\Delta B)\cap (A\Delta C) </math>

Symmetric differences are used to create filters for forcing constructions.