Let A and B be the sets. The symmetric difference between A and B is :
\(A \oplus B = (A - B) \cup (B - A)\)
The symmetric difference of two sets contains the elements that belong to exactly one of the two sets, that is, to either set but not to both.
The elements that are in A but not in B form \(A - B\), and the elements that are in B but not in A form \(B - A\).
Taking the union of these two disjoint parts gives \(A \oplus B = (A - B) \cup (B - A)\), which can also be written as \((A \cup B) - (A \cap B)\).
The remaining relations do not define the symmetric difference: they put the union, the intersection or the subset relation on the left-hand side and are not valid set identities.
Hence, \(A \oplus B = (A - B) \cup (B - A)\).
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