The Klein four-group is also isomorphic to the direct sum , so that it can be represented as the pairs under component-wise addition modulo 2 (or equivalently the bit strings under bitwise XOR); with (0,0) being the group's identity element. The Klein four-group is thus an example of an elementary abelian 2-group, which is also called a Boolean group. 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. formula_2; the empty set is the group's identity element in this case.