Distributive property

Visualization of distributive law for positive numbers

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from Boolean algebra and elementary algebra. In propositional logic, distribution refers to two valid rules of replacement. The rules allow one to reformulate conjunctions and disjunctions within logical proofs.

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), multiplication by 2 is said to distribute over addition of 1 and 3.Since one could have put any real numbers in place of 2, 1, and 3 above, and still have obtained a true equation, multiplication of real numbers distributes over addition of real numbers.

Definition

Given a set S and two binary operators ∗ and + on S, the operation:

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

${\displaystyle x*(y+z)=(x*y)+(x*z),}$

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

${\displaystyle (y+z)*x=(y*x)+(z*x),}$ and

∗ is distributive over + if it is left- and right-distributive.[1]

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

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