Distributive property

Distribution is a concept from algebra: It tells how binary operations are to be handled. The most simple case is that of addition and multiplication of numbers. 
For example, in arithmetic:
 2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3).

In the left-hand  side of the first equation, the 2 multiplies the sum of 1 and 3; on the right-hand side, it multiplies the 1 and the 3 individually, with the products added afterwards.
Because these give the same final answer (8), it is said that multiplication by 2 distributes over addition of 1 and 3.
Since one could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, we say that multiplication of real numbers distributes over addition of real numbers.

Definition

Given a set  and two binary operators ∗ and + on , we say that the operation:

∗ is left-distributive over + if, given any elements , and  of ,

∗ is right-distributive over + if, given any elements , and  of ,
 and
∗ is distributive over + if it is left- and right-distributive.Ayres, p. 20
Notice that when ∗ is commutative, the three conditions above are logically equivalent.

 Applications 
The distributive property can also be applied to:
 Real numbers
 Complex numbers
 Matrices (special rules apply)
 Vectors (special rules apply)
 Sets
Propositional logic

References
 Ayres, Frank, Schaum's Outline of Modern Abstract Algebra, McGraw-Hill; 1st edition (June 1, 1965). .

Category:Algebra
Category:Rules of inference