Distributivity

Distributivity is a property of two binary operations, such as addition and multiplication. For multiplication and addition, distributivity tells us that we can “expand the brackets”. For example, given the expression

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

we know that we can expand the brackets, which gives a different expression

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

Both expressions lead to the same result, 14. We could expand the brackets because multiplication on the set of real numbers is distributive over addition. In general, if we have two operations · and +, we say that · is distributive over + on a set M if

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

for all a, b, c ∈ M. Other examples of distributive operations are the operations of propositional logic: conjunction ∧ and disjunction ∨. Suppose we have statements A, B and C. If we say “A and (B or C)”, we write it as

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

and because conjunction is distributive over disjunction, we also have

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

In words: “(A and B) or (A and C)”. To have an example of operations that are not distributive, take subtraction and addition:

$$10-(5+3)$$

The correct result is of course 2, because after adding up the bracket we get 10 − 8, which is 2. If we applied the distributive law to this calculation, we would get the expression:

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

After working out the brackets we would have 5 + 7, which is 12. We can see that we got the wrong result, so subtraction is not distributive over addition.

More examples: some operations with matrices satisfy the distributive law. So do the vectors in a vector space. The operations in a field must be distributive.