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March 26, 2010 / porton

Erroneous theorem

I found a counter-example and an error in my proof of this (erroneous) theorem in Funcoids and Reloids article:

Let f\in\mathsf{FCD}. If R is a set of co-complete funcoids then f \circ \bigcup {\nobreak}^{\mathsf{FCD}} R = \bigcup {\nobreak}^{\mathsf{FCD}} \left\langle f \circ \right\rangle R.

A counter-example: Let \Delta = \{ (-\epsilon;\epsilon) | \epsilon\in\mathbb{R}, \epsilon>0 \}. Let f = \Delta \times^{\mathsf{FCD}} \mathbb{R} and R = \{ \mathbb{R}\times(\epsilon;+\infty) | \epsilon\in\mathbb{R}, \epsilon>0 \}.

Then \bigcup {\nobreak}^{\mathsf{FCD}} R = \bigcup R = \mathbb{R} \times (0;+\infty); f \circ \bigcup {\nobreak}^{\mathsf{FCD}} R = \mathbb{R} \times \mathbb{R}; \bigcup {\nobreak}^{\mathsf{FCD}} \langle f \circ \rangle R = \bigcup {\nobreak}^{\mathsf{FCD}} \{ \emptyset \} = \emptyset.

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