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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 tex2html_wrap_inline32625 defined by:

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This mapping is not an isomorphism, although it allows one to identify the floating point numbers with tex2html_wrap_inline32627 . The mapping tex2html_wrap_inline32629 defined by:

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is an isomorphism between tex2html_wrap_inline32627 , a subset of the intervals, and tex2html_wrap_inline32371 . A mapping tex2html_wrap_inline32635 is an isomorphism if tex2html_wrap_inline32637 is a homomorphism from tex2html_wrap_inline32639 to tex2html_wrap_inline32641 , and tex2html_wrap_inline32643 is a homomorphism from tex2html_wrap_inline32641 to tex2html_wrap_inline32639 .

Addition inherits the identity tex2html_wrap_inline32649 while multiplication inherits the identity tex2html_wrap_inline32651 . Since intervals were constructed with mathematical rigor in mind, several nice properties are obeyed by intervals. Chief among these is the sub-distributive law:

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

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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 up previous notation contents
Next: 2.8 Real Interval Arithmetic Up: 2.7 Interval Arithmetic Previous: 2.7.4 Interval Extension
Jeff TupperMarch 1996