For sets A and B (subsets of some universal set), the symmetric difference, denoted by A+B, is the set (A∖B) ∪ (B∖A). The notation AΔB is also used. Its logical equivalent is exclusive disjunction or ‘xor’. The set is represented by the shaded regions of the Venn diagram shown. The following properties hold, for all A, B, and C (subsets of some universal set E):
(i) A+A = Ø, A+Ø = A, A+A′ = E, A+E = A′.
(ii) A+B = (A ∪ B)∖(A ∩ B)
= (A ∪ B) ∩ (A′ ∪ B′).
(iii) A+B = B+A, the commutative law.
(iv) (A+B)+C = A+(B+C), the associative law.
(v) A ∩ (B+C) = (A ∩ B)+(A ∩ C), the operation ∩ is distributive over the operation+.
The symmetric difference AΔB
See set difference.
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