Distributive Operation Relation

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]

• Example(s):
  • Multiplication Distributive Law.
    • (Multiplication, Addition)
    • 2×(1+3) = (2×1) + (2×3).
    • a×(b+c) = (a×b) + (a×c).
    • α(v+w) = αv + αw.
  • Division 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.

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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]
  • Notice that when · is commutative, then the three above conditions are logically equivalent.

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