Multiplication Distributive Law

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

• AKA: Distributive Law.
  • It ranges from being a left-distributive to being right-distributive operation:
    • [math]\displaystyle{ x \times (y \pm z) = (x \times y) \pm (x \times z) }[/math]
    • [math]\displaystyle{ (y \pm z) \times x = (y \times x) \pm (z \times x) }[/math] (right multiplication distribution)

      both left and right distributive expressions are logically equivalent as multiplication is commutative.

• Example(s):
  • Let's consider [math]\displaystyle{ x = 2 }[/math], [math]\displaystyle{ y=3 }[/math], [math]\displaystyle{ z=5 }[/math].
    • Distributivity of multiplication over addition:
      • [math]\displaystyle{ x \times (y + z)=2\times(3+5)=2\times 8=16 \iff (x \times y) + (x \times z)= (2\times 3)+ (2\times 5)= 6+10 =16 }[/math]
      • [math]\displaystyle{ (y + z) \times x = (3\times 5)\times2= 8\times 2 =16 \iff (y \times x) + (z \times x)= (3\times 2)+(5\times 2)= 6+10=16 }[/math]
    • Distributivity of multiplication over subtraction:
      • [math]\displaystyle{ x \times (y - z) = 2 \times (3 - 5)= 2\times 2=4 \iff (x \times y) - (x \times z) (2 \times 3) - (2 \times 5)=6 -10=4 }[/math]
      • [math]\displaystyle{ (y - z) \times x =(3 - 5) \times 2=2\times 2=4 \iff (y \times x) - (z \times x)=(3 \times 2) - (5 \times 2)=6-10=4 }[/math]
• Counter-Example(s)
  • Set Distributive Law.
  • Division 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.

(...) In either case, the distributive property can be described in words as: To multiply a sum (or difference) by a factor, each summand (or minuend and subtrahend) is multiplied by this factor and the resulting products are added (or subtracted).

If the operation outside the parentheses (in this case, the multiplication) is commutative, then left-distributivity implies right-distributivity and vice versa.

2017

• (MathWorld, 2017) ⇒ Weisstein, Eric W. “Distributive." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Distributive.htm
  • Distributive: A multiplication * is said to be right distributive if

[math]\displaystyle{ (x+y)z=xz+yz }[/math]

for every [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], and [math]\displaystyle{ z }[/math]. Similarly, it is said to be left distributive if

[math]\displaystyle{ z(x+y)=zx+zy }[/math]

for every [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], and [math]\displaystyle{ z }[/math].

If a multiplication is both right- and left-distributive, it is simply said to be distributive. For example, the real numbers R are distributive.

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