A Calculus of Number Based on Spatial Forms - University of ...

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Inversion: A =
:
Equivalent Expressions Template Replacement
A =
([])
([][])
A = []
()
()
A = ()
A =
Figure 5.1: Examples <strong>of</strong> Inversion.
5.3.1 Properties <strong>of</strong> the Additive Inverse
The inverse boundary can be manipulated in ways analogous the the additive inverse.
The basic properties <strong>of</strong> the inverse are outlined by three theorems: inverse
collection, inverse cancellation, and inverse promotion. The theorem <strong>of</strong> inverse collection
transforms collected inversions into a single inversion and the theorem <strong>of</strong>
inverse cancellation removes pairs <strong>of</strong> nested inverses.
Theorem 1 (Inverse Collection) Acollection <strong>of</strong> inverted elements equals an inversion
<strong>of</strong> the collection, as
=:
A pro<strong>of</strong> <strong>of</strong> inverse collection follows from inversion:
=AB=:
Theorem 2 (Inverse Cancellation) The inverse boundary is its own functional
inverse, as
=A: