However, commutativity does not imply associativity. This allows an easier use of associativity and commutativity. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The nontrivial part is a proof of Poisson commutativity of these functions. Functors are represented by arrows between categories, subject to specific defining commutativity conditions. The quaternion involved abandoning commutativity, a radical step for the time. The difference of two squares identity is proved using the commutativity of multiplication. This is essentially equivalent to the commutativity of multiplication. A second difficulty occurs with the commutativity of addition and multiplication. The involutivity condition is a generalization of the commutativity of partial derivatives. Some forms of symmetry can be directly linked to commutativity. More general conditions which guarantee commutativity of a ring are also known. Commutativity&colon; Addition and multiplication of real numbers are both commutative. See, for example, the proof of commutativity accompanying "addition of natural numbers".