This implies 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 .)
42.
Taken together, we see that the power set of any set " X " becomes an abelian group if we use the symmetric difference as operation . ( More generally, any field of sets forms a group with the symmetric difference as operation . ) A group in which every element is its own inverse ( or, equivalently, in which every element has order 2 ) is sometimes called a Boolean group; the symmetric difference provides a prototypical example of such groups.
43.
Taken together, we see that the power set of any set " X " becomes an abelian group if we use the symmetric difference as operation . ( More generally, any field of sets forms a group with the symmetric difference as operation . ) A group in which every element is its own inverse ( or, equivalently, in which every element has order 2 ) is sometimes called a Boolean group; the symmetric difference provides a prototypical example of such groups.
44.
Taken together, we see that the power set of any set " X " becomes an abelian group if we use the symmetric difference as operation . ( More generally, any field of sets forms a group with the symmetric difference as operation . ) A group in which every element is its own inverse ( or, equivalently, in which every element has order 2 ) is sometimes called a Boolean group; the symmetric difference provides a prototypical example of such groups.
45.
The Klein four-group is thus also the group generated by the symmetric difference as the binary operation on the subsets of a powerset of a set with two elements, i . e . over a field of sets with four elements, e . g . \ { \ emptyset, \ { \ alpha \ }, \ { \ beta \ }, \ { \ alpha, \ beta \ } \ }; the empty set is the group's identity element in this case.
