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# symmetric difference

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For sets A and B (subsets of some universal set), the symmetric difference, denoted by A+B, is the set (AB) ∪ (BA). The notation A Δ B is also used. The set is represented by the shaded regions of the Venn diagram shown below. 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=(AB)∖(AB)=(AB) ∩ (A′ ∪ B′).(iii)A+B=B+A, the commutative law.(iv) (A+B)+C=A+(B+C), the associative law.(v)A ∩ (B+C)=(AB)+(A ∩ C), the operation ∩ is distributive over the operation+.

(i)A+A=Ø, A+Ø=A, A+A′=E, A+E=A′.

(ii)A+B=(AB)∖(AB)=(AB) ∩ (A′ ∪ B′).

(iii)A+B=B+A, the commutative law.

(iv) (A+B)+C=A+(B+C), the associative law.

(v)A ∩ (B+C)=(AB)+(A ∩ C), the operation ∩ is distributive over the operation+.

Subjects: Mathematics.

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