symmetricdifference

Symmetric difference, denoted A Δ B, is a binary operation on sets defined by A Δ B = (A ∪ B) − (A ∩ B). Equivalently, A Δ B = (A \ B) ∪ (B \ A). It consists of the elements that belong to exactly one of the sets.

Properties: It is commutative (A Δ B = B Δ A) and associative ((A Δ B) Δ C = A Δ (B Δ C)). The empty set is the identity: A Δ ∅ = A. Each set is its own inverse under Δ: A Δ A = ∅. If A = B, the difference is empty.

Interpretation: In terms of characteristic functions, χ_{A Δ B} = χ_A ⊕ χ_B, i.e., the XOR of membership. It also coincides with bitwise XOR for finite bitstrings, making it useful in computer science.

Generalization: For a finite collection of sets, the symmetric difference is obtained by iteratively applying Δ, and its result is the set of elements that belong to an odd number of the sets.

Example: A = {1, 2, 3}, B = {2, 3, 4}. A Δ B = {1, 4}.

Applications: used in set theory to study parity, in algorithms for parity checks, in graph theory for edge duplication cancellation, and in boolean algebra as XOR.

See also: union, intersection, difference, XOR, Boolean algebra, power set.