Distributive property
Given two binary operations and
acting on a set
, we say that
has the distributive property over
(or
distributes over
) if, for all
we have
and
.
Note that if the operation is commutative, these two conditions are the same, but if
does not commute then we could have operations which left-distribute but do not right-distribute, or vice-versa.
Also note that there is no particular reason that distributivity should be one-way, as it is with conventional multiplication and addition. For example, the set operations union () and intersection (
) distribute over each other: for any sets
we have
and
.
(In fact, this is a special case of a more general setting: in a distributive lattice, each of the operations meet and join distributes over the other. Meet and join correspond to union and intersection when the lattice is a boolean lattice.)
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