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 .
Key Note - This isn't an example of the Distributive Property! This is actually using the Associative Property, NOT THE DISTRIBUTIVE PROPERTY!
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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