Distributive property

property involving two mathematical operations

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

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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.[1] Notice that when ∗ is commutative, the three conditions above are logically equivalent.

Applications

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The distributive property can also be applied to:

References

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  • Ayres, Frank, Schaum's Outline of Modern Abstract Algebra, McGraw-Hill; 1st edition (June 1, 1965). ISBN 0-07-002655-6.
  1. Ayres, p. 20