# Distributive Operation Relation

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A Distributive Operation Relation is an equality relation between two arithmetic operations of that is based on the distributive law.

**AKA:**Distributive Operation, Distributivity Operation, Distributivity Property.**Context:**- It ranges from being a Left-Distributive Operation to being Right-Distributive Operation:
- Left Distributive Operation : [math]\displaystyle{ x\star(y\odot z)=(x\star y) \odot (x \star z) }[/math].
- Right Distributive Operation: [math]\displaystyle{ (y\odot z)\star x=(y\star x) \odot (z \star x) }[/math].
Both Operation Relations define the distributivity of [math]\displaystyle{ \star }[/math] over a [math]\displaystyle{ \odot }[/math]

- It ranges from being a Left-Distributive Operation to being Right-Distributive Operation:
**Example(s):**- Multiplication Distributive Law.
- (Multiplication, Addition)
- 2×(1+3) = (2×1) + (2×3).
`a×(b+c) = (a×b) + (a×c)`

.*α*(**v**+**w**) =*α*v**+**w.*α*

- Division Distributive Law.

- Multiplication Distributive Law.
**Counter-Example(s):**`(5+2)+1=5+(2+1)`

, application of an Associative Operation.`(5+2)+1=(2+5)+1`

, application of a Commutative Operation.

**See:**Nondistributive Operation, Distributivity Axiom, Commutative Operation, Associative Operation.

## References

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Distributivity
- In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example:
`2 x (1 + 3) = (2 x 1) + (2 x 3)`

.

- In the left-hand side of the above 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 results added afterwards. Because these give the same f==Definition==
- Given a set S and two binary operations · and + on S, we say that the operation ·
- is left-distributive over + if, given any elements x, y, and z of S:
`x · (y + z) = (x · y) + (x · z)`

; - is right-distributive over + if, given any elements x, y, and z of S:
`(y + z) · x = (y · x) + (z · x)`

; - is distributive over + if it is both left- and right-distributive. [1]

- is left-distributive over + if, given any elements x, y, and z of S:
- Notice that when · is commutative, then the three above conditions are logically equivalent.

- In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: