Symmetric difference Guide, Meaning , Facts, Information and Description
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
AΔ
B.
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 symmetic difference is equivalent to the union of both relative complementss, 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
singletonss 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.
See also
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