symmetric difference

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

For example, the symmetric difference of the sets and is .

Symmetric operators on sets include the union, intersection, and symmetric difference.

A state is a set of levels whose symmetric difference with the vacuum state is finite.

(But note that for the diameter of the symmetric difference the triangle inequality does not hold.)

The symmetric difference is equivalent to the union of both relative complements, that is:

For every pair of circuits of the matroid, their symmetric difference contains another circuit.

Taken together, we see that the power set of any set X becomes an abelian group if we use the symmetric difference as operation.

Further properties of the symmetric difference:

Symmetric difference of sets.

The following condition is not sufficient for to be generating: for every there exists a Borel set such that ( means symmetric difference).

The symmetric difference is commutative and associative:

Symmetric difference is measurable: .

The distance between two sets is defined as the measure of the symmetric difference of the two sets.

For large , the symmetric difference has size , while has size .

Later authors changed the interpretation, commonly reading it as exclusive or, or in set theory terms symmetric difference; this step means that addition is always defined.

Overlay functions (including intersection, difference, union, symmetric difference)

Intersection distributes over symmetric difference:

An elementary observation is that one cannot have exactly 2 subgroups of index 2, as the complement of their symmetric difference yields a third.

Note that the symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric.

In geometry, a double wedge is the (closure of) the symmetric difference of two half-spaces whose boundaries are not parallel to each other.

A symmetric difference overlay defines an output area that includes the total area of both inputs except for the overlapping area.

To correct first find , then find a set v where , finally compute the symmetric difference to get .

Conversely, suppose that a matching is not optimal, and let be the symmetric difference where is an optimal matching.

As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations.