Let the fibre of $x$ over some binary relation $R$ be the set of $y$ such that $xRy$. Question: Is the following claim true?
$\forall R \forall X \exists x$ the fibre of $x$ over $R$ is $X$.
Question about fibres
-1
$\begingroup$
elementary-set-theory
-
0The claim in the box is obviously false. Are you sure you wrote down the question correctly? – 2017-01-22
1 Answers
2
As stated, $X$ may not have anything to do with $R$, so as stated this is immediately not true (e.g. $R$ may be about numbers , but $X$ could be a set of fruits)
It would be a little more interesting if we say that $R$ is defined over $X$, i.e. if $R \subseteq X \times X$
But even then, the statement is not true. Counterexample:
$X = \{ 1 \}$ (anything not empty)
$R = \{ \}$