Distributivity

Distributivity is a property of two binary operations, such as addition and multiplication. Distributivity, in the case of multiplication and addition, tells us that we can "multiply the parenthesis". For example, if we have an expression

$$2\cdot(3+4)$$

then we know that we can multiply the parentheses to get a different expression

$$2\cdot(3+4) = 2\cdot3+2\cdot4$$

Both expressions lead to the same result, the number 14. We could do this multiplication because the multiplication operation is distributive on the set of real numbers with respect to the addition operation. Thus, in general, if we have two operations · and +, then we say that the operation · is distributive on the set M with respect to the operation +, if

$$\begin{eqnarray} a\cdot(b+c) &=& (a\cdot b) + (a\cdot c)\\ (b+c)\cdot a &=& (b\cdot a) + (c\cdot a) \end{eqnarray}$$

for all a, b, c ∈ M. Another example of distributive operations are operations from propositional logic, conjunction ∧ and disjunction ∨. Suppose we have the propositions A, B, and C. If we say "A and at the same time (B or C)", we write this as

$$A \wedge (B \vee C)$$

And since the operation of conjunction is distributive with respect to the operation of disjunction, the simultaneous

$$(A \wedge B) \vee (A \wedge C)$$

Thus, in words, "(A and at the same time B) or (A and at the same time C)". To have an example of operations that are not distributive, let's take addition and subtraction:

$$10-(5+3)$$

The correct result is obviously 2, because after adding the parentheses we get 10 − 8, which is 2. If we applied the distributivity rule to this calculation, we would get the expression:

$$(10-5)+(10-3)$$

After adding the parentheses, we would have 5 + 7, which is 12. We can see that we got the wrong result, so the addition and subtraction operations are not distributive.

Other examples: some matrix operations satisfy the distributive law. So do vectors in vector space. Operations on a solid must be distributive.