Symmetric difference

In mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets, but not in both. This operation is the set-theoretic kin of the exclusive disjunction (XOR operation) in Boolean logic. The symmetric difference of the sets A and B is commonly denoted by

Failed to parse (Missing texvc executable; please see math/README to configure.): A \Delta 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 symmetric difference is equivalent to the union of both relative complements, that is:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \Delta B = (A - B) \cup (B - A),\,

and it can also be expressed as the union of the two sets, minus their intersection:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \Delta B = (A \cup B) - (A \cap B),

or with the XOR operation:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \Delta B = \{x : (x \in A) \mbox{ XOR } (x \in B)\}.

The symmetric difference is commutative and associative:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \Delta B = B \Delta A,\,

Failed to parse (Missing texvc executable; please see math/README to configure.): (A \Delta B) \Delta C = A \Delta (B \Delta C).\,

Thus, the repeated symmetric difference is an operation on a multiset of sets giving the set of elements which are in an odd number of sets.

The symmetric difference of two repeated symmetric differences is the repeated symmetric difference of the join of the two multisets, where for each double set both can be removed. In particular:

Failed to parse (Missing texvc executable; please see math/README to configure.): (A \Delta B) \Delta (B \Delta C) = A \Delta C.\,

This implies a kind of triangle inequality: the symmetric difference of A and C is contained in the union of the symmetric difference of A and B and that of B and C. (But note that for the diameter of the symmetric difference the triangle inequality does not hold.)

The empty set is neutral, and every set is its own inverse:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \Delta \varnothing = A,\,

Failed to parse (Missing texvc executable; please see math/README to configure.): A \Delta A = \varnothing.\,

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \cap (B \Delta C) = (A \cap B) \Delta (A \cap 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

Failed to parse (Missing texvc executable; please see math/README to configure.): x \Delta y = (x \lor y) \land \lnot(x \land y) = (x \land \lnot y) \lor (y \land \lnot x) = x \oplus y.

This operation has the same properties as the symmetric difference of sets.

n-ary symmetric difference

As above, the symmetric difference of a collection of sets contains just elements which are in an odd number of the sets in the collection:

Failed to parse (Missing texvc executable; please see math/README to configure.): \triangle M = \left\{ a \in \bigcup M: |\{A\in M|a \in A\}| \mbox{ is odd}\right\}

. Evidently, this is well-defined only when each element of the union Failed to parse (Missing texvc executable; please see math/README to configure.): \bigcup M

is contributed by a finite number of elements of Failed to parse (Missing texvc executable; please see math/README to configure.): M

.

Symmetric difference on measure spaces

As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are. Formally, if μ is a σ-finite measure defined on a σ-algebra Σ, the function,

Failed to parse (Missing texvc executable; please see math/README to configure.): d(X,Y) = \mu(X \Delta Y)

is a pseudometric on Σ. d becomes a metric if Σ is considered modulo the equivalence relation X ~ Y if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): \mu(X \Delta Y) = 0 . The resulting metric space is separable if and only if L²(μ) is separable.

See also

de:Mengenlehre#Differenz_und_Komplement

it:Differenza simmetrica he:הפרש סימטרי ja:対称差 pl:Różnica symetryczna ru:Симметрическая разность uk:Симетрична різниця множин