Distributive Property
If a set is closed under two or more binary operations (*Ө) for all a,b and c € S, such that:

a*(bӨc) = (a*b)Ө(a*c) - Left distributive

(BӨc)*a = (b*a)Ө(c*a) - Right distributive over the operation Ө

Example: Given the set R of real numbers under the operations * and Ө defined by: a*b = a + b - 3, aӨb = 5ab for all a, b € R. Does * distribute over Ө.

Solution:

Let a, b, c € R

a * (bӨc) = (a*b) Ө (a*c)

a * (bӨc) = a*(5ab)

= a + 5ab - 3

(a*b) Ө (a*c) = (a+b-3) Ө (a+c-3)

=5(a+b-3)(a+c-3)

From the expansion, it's obvious that a*(bӨc) ≠ (a*b) Ө (a*c) therefore * does not distribute over