Next: 2.8 Real Interval Arithmetic Up: 2.7 Interval Arithmetic Previous: 2.7.4 Interval Extension

## 2.7.5 Algebraic Properties

Intervals generalize floating point numbers since there is an injective mapping defined by:

This mapping is not an isomorphism, although it allows one to identify the floating point numbers with . The mapping defined by:

is an isomorphism between , a subset of the intervals, and . A mapping is an isomorphism if is a homomorphism from to , and is a homomorphism from to .

Addition inherits the identity while multiplication inherits the identity . Since intervals were constructed with mathematical rigor in mind, several nice properties are obeyed by intervals. Chief among these is the sub-distributive law:

Although neither addition nor multiplication are associative, the operators preserve ``associative trails''. This property is expressed, for addition, as follows:

The property follows from the associativity of real addition and the inclusion property of interval addition. The above property can be extended by considering that interval addition is commutative. In general, a real computation result is guaranteed to be contained in the result of the associated interval computation because the interval inclusion property is transitive.

Next: 2.8 Real Interval Arithmetic Up: 2.7 Interval Arithmetic Previous: 2.7.4 Interval Extension
 Jeff Tupper March 1996