Real numbers In the following examples, the use of the distributive law on the set of real numbers \R is illustrated. When multiplication is mentioned in elementary mathematics, it usually refers to this kind of multiplication. From the point of view of algebra, the real numbers form a
field, which ensures the validity of the distributive law. {{defn| \begin{align} (a + b) \cdot (a - b) & = a \cdot (a - b) + b \cdot (a - b) = a^2 - ab + ba - b^2 = a^2 - b^2 \\ & = (a + b) \cdot a - (a + b) \cdot b = a^2 + ba - ab - b^2 = a^2 - b^2 \\ \end{align} Here the distributive law was applied twice, and it does not matter which bracket is first multiplied out. }}
Matrices The distributive law is valid for
matrix multiplication. More precisely, (A + B) \cdot C = A \cdot C + B \cdot C for all l \times m-matrices A, B and m \times n-matrices C, as well as A \cdot (B + C) = A \cdot B + A \cdot C for all l \times m-matrices A and m \times n-matrices B, C. Because the commutative property does not hold for matrix multiplication, the second law does not follow from the first law. In this case, they are two different laws.
Other examples •
Multiplication of
ordinal numbers, in contrast, is only left-distributive, not right-distributive. • The
cross product is left- and right-distributive over
vector addition, though not commutative. • For
sets, the
union is distributive over
intersection, and intersection is distributive over union. •
Logical disjunction ("or") is distributive over
logical conjunction ("and"), and vice versa. • For
real numbers (and for any
totally ordered set), the
maximum operation is distributive over the
minimum operation, and vice versa: \max(a, \min(b, c)) = \min(\max(a, b), \max(a, c)) \quad \text{ and } \quad \min(a, \max(b, c)) = \max(\min(a, b), \min(a, c)). • For
integers, the
greatest common divisor is distributive over the
least common multiple, and vice versa: \gcd(a, \operatorname{lcm}(b, c)) = \operatorname{lcm}(\gcd(a, b), \gcd(a, c)) \quad \text{ and } \quad \operatorname{lcm}(a, \gcd(b, c)) = \gcd(\operatorname{lcm}(a, b), \operatorname{lcm}(a, c)). • For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + \max(b, c) = \max(a + b, a + c) \quad \text{ and } \quad a + \min(b, c) = \min(a + b, a + c). • For
binomial multiplication, distribution is sometimes referred to as the
FOIL Method (First terms a c, Outer a d, Inner b c, and Last b d) such as: (a + b) \cdot (c + d) = a c + a d + b c + b d. • In all
semirings, including the
complex numbers, the
quaternions,
polynomials, and
matrices, multiplication distributes over addition: u (v + w) = u v + u w, (u + v)w = u w + v w. • In all
algebras over a field, including the
octonions and other
non-associative algebras, multiplication distributes over addition. == Propositional logic ==