Distributive Property - Examples

Examples

1. Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers.
2. Multiplication of ordinal numbers, in contrast, is only left-distributive, not right-distributive.
3. The cross product is left- and right-distributive over vector addition, though not commutative.
4. Matrix multiplication is distributive over matrix addition, though also not commutative.
5. The union of sets is distributive over intersection, and intersection is distributive over union.
6. Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction.
7. For real numbers (or 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)) and min(a,max(b,c)) = max(min(a,b),min(a,c)).
8. For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd(a,lcm(b,c)) = lcm(gcd(a,b),gcd(a,c)) and lcm(a,gcd(b,c)) = gcd(lcm(a,b),LCM(a,c)).
9. For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) = min(a+b,a+c).