What is Distributive property?
'''Distribution''' is a concept from [[algebra]]: It tells how [[binary operation]]s are to be handled. The most simple case is that of [[addition]] and [[multiplication]] of [[numbers]]. For example, in [[arithmetic]]: : 2 ⋅ (1 + 3) = (2 ⋅ 1) + (2 ⋅ 3), but 2 / (1 + 3) ≠ (2 / 1) + (2 / 3). In the left-hand side of the first 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 products added afterwards. Because these give the same final answer (8), it is said that multiplication by 2 ''distributes'' over addition of 1 and 3. Since one could have put any [[real number]]s in place of 2, 1, and 3 above, and still have obtained a true equation, we say that [[multiplication]] of real numbers ''distributes'' over [[addition]] of real numbers. ==Definition== Given a [[Set (mathematics)|set]] <math>S</math> and two binary operators ∗ and + on <math>S</math>, we say that the operation: ∗ is ''left-distributive'' over + if, [[given any]] elements <math>x,y</math>, and <math>z</math> of <math>S</math>, ::<math>x*(y+z)=(x*y)+(x*z),</math> ∗ is ''right-distributive'' over + if, given any elements <math>x,y</math>, and <math>z</math> of <math>S</math>, ::<math>(y+z)*x=(y*x)+(z*x),</math> and ∗ is ''distributive'' over + if it is left- and right-distributive.<ref>Ayres, [https://books.google.com/books?id=7r3bVWx2GHkC&pg=PA20&dq=%22binary+operation%22+%22Left+distributive%22#PPA20,M1 p. 20]</ref> Notice that when ∗ is [[commutative]], the three conditions above are [[logical equivalence|logically equivalent]]. == Applications == The distributive property can also be applied to: * [[Real number]]s * [[Complex number]]s * [[Matrix (mathematics)|Matrices]] (special rules apply) * [[Vector]]s (special rules apply) * [[Set]]s *[[Propositional logic]] ==References== * Ayres, Frank, ''Schaum's Outline of Modern Abstract Algebra'', McGraw-Hill; 1st edition (June 1, 1965). {{ISBN|0-07-002655-6}}. {{reflist}} [[Category:Algebra]] [[Category:Rules of inference