Symmetric difference
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In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. This operation is the set-theoretic equivalent of the XOR operation in Boolean logic. The symmetric difference of the sets A and B is commonly denoted by AΔB.
For example, the symmetric difference of the sets {1,2,3} and {3,4} is {1,2,4}. The symmetric difference of the set of all students and the set of all females consists of all male students together with all female non-students.
The symmetric difference is equivalent to the union of both relative complements, that is:
- A Δ B = (A − B) ∪(B − A)
and it can also be expressed as the union of the two sets, minus their intersection:
- A Δ B = (A ∪B) − (A ∩B)
or with the XOR operation:
- A Δ B = { x : (x ∈A) XOR (x ∈B) }.
The symmetric difference is commutative and associative:
- A Δ B = B Δ A
- (A Δ B) Δ C = A Δ (B Δ C)
The empty set is neutral, and every set is its own inverse:
- A Δ Ø = A
- A Δ A = Ø
Taken together, we see that the power set of any set X becomes an abelian group if we use the symmetric difference as operation. Because every element in this group is its own inverse, this is in fact a vector space over the field with 2 elements Z2. If X is finite, then the singletons form a basis of this vector space, and its dimension is therefore equal to the number of elements of X. This construction is used in graph theory, to define the cycle space of a graph.
Intersection distributes over symmetric difference:
- A ∩(B Δ C) = (A ∩B) Δ (A ∩C)
and this shows that the power set of X becomes a ring with symmetric difference as addition and intersection as multiplication. This is the prototypical example of a Boolean ring.
The symmetric difference can be defined in any Boolean algebra, by writing
- x Δ y = (x ∨ y) ∧ ¬(x ∧ y) = (x ∧ ¬y) ∨ (y ∧ ¬x)
This operation has the same properties as the symmetric difference of sets.