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 ${\displaystyle S}$ and two binary operators ∗ and + on ${\displaystyle S}$, we say that the operation:

∗ is left-distributive over + if, given any elements ${\displaystyle x,y}$, and ${\displaystyle z}$ of ${\displaystyle S}$,

${\displaystyle x*(y+z)=(x*y)+(x*z),}$

∗ is right-distributive over + if, given any elements ${\displaystyle x,y}$, and ${\displaystyle z}$ of ${\displaystyle S}$,

${\displaystyle (y+z)*x=(y*x)+(z*x),}$ and

∗ is distributive over + if it is left- and right-distributive. 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). ISBN 0-07-002655-6.