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

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

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

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

## 1 thought on “Erroneous theorem”