Division Distributive Law

An Division Distributive Law is a distributive operation relation between division operation ([math]\displaystyle{ / }[/math]) and addition ([math]\displaystyle{ + }[/math]) or subtraction ([math]\displaystyle{ - }[/math]) binary operations.

• AKA: Distributive Law.
  • It is a right-distributive operation and can be defined as
    • [math]\displaystyle{ (x \pm y) / z = (x / z) \pm (y / z) }[/math]

      Note that division is can not a right-distributive operation because is not commutative.

• Example(s):
  • Let's consider [math]\displaystyle{ x = 7 }[/math], [math]\displaystyle{ y=3 }[/math], [math]\displaystyle{ z=5 }[/math].
    • Distributivity of division over addition:
  • [math]\displaystyle{ (x + y) / z= (7+3)/5 = 10/5 = 2 \iff x / z + y / z = (7/5) + (3/5)=10/5=2 }[/math]
    • Distributivity of division over subtraction:
  • [math]\displaystyle{ (x - y) / z= (7-3) \div 5 = 4/5 \iff (x / z) - (y / z) = 7/5 - 3/5=4/5 }[/math]
• Counter-Example(s)
  • Let's consider [math]\displaystyle{ x = 7 }[/math], [math]\displaystyle{ y=3 }[/math], [math]\displaystyle{ z=5 }[/math].
    • [math]\displaystyle{ x/(y + z) = 7/(3+5)=7/8 \neq (x / y) + (x/z)=(7/3)+(7/5)=(35/15)+(21/15)=56/15 }[/math].
    • [math]\displaystyle{ x/(y - z) = 7/(3-5)=7/2 \neq (x / y) - (x/z)=(7/3)-(7/5)=(35/15)-(21/15)=14/15 }[/math].
  • Set Distributive Law.
  • Multiplication Distributive Law.
• See: Distributive Operation Relation, Venn Diagram, Set Theory, Linear Logic, Binary Operation, Union (Set Theory), Intersection (Set Theory), Identity (Mathematics), Commutative Operation, Associativity, Distributivity, Empty Set, Universal Set, Complement (Set Theory), Identity Element.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Distributive_property#Definition Retrieved:2017-6-25.
  • Given a set S and two binary operators ∗ and + on S, we say that the operation:

∗ is left-distributive over + if, given any elements x, y, and z of S,

[math]\displaystyle{ x * (y + z) = (x * y) + (x * z), }[/math]

∗ is right-distributive over + if, given any elements x, y, and z of S,

[math]\displaystyle{ (y + z) * x = (y * x) + (z * x), }[/math] and

∗ is distributive over + if it is left- and right-distributive.

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

(...) One example of an operation that is "only" right-distributive is division, which is not commutative:

[math]\displaystyle{ (a \pm b) \div c = a \div c \pm b \div c }[/math]

In this case, left-distributivity does not apply:

[math]\displaystyle{ a \div(b \pm c) \neq a \div b \pm a \div c }[/math]

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