Distributive property

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Given two binary operations $\times$ and $+$ acting on a set $S$, we say that $\times$ has the distributive property over $+$ (or $\times$ distributes over $+$) if, for all $a, b, c \in S$ we have $a\times(b + c) = (a\times b) + (a \times c)$ and $(a + b) \times c = (a \times c) + (b \times c)$.

Note that if the operation $\times$ is commutative, these two conditions are the same, but if $\times$ does not commute then we could have operations which left-distribute but do not right-distribute, or vice-versa.

Key Note - This isn't an example of the Distributive Property! \[a(b \times c) = ab \times ac.\] 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 ($\cup$) and intersection ($\cap$) distribute over each other: for any sets $A, B, C$ we have $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap(B \cup C) = (A \cap B) \cup (A \cap C)$.

(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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