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What is the negation of the following case:

Case A: The function $f$ has a unique rational fixed point $a$.

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3
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I would say the function has:

  • multiple rational fixed points or
  • no fixed point at all or
  • at least one irrational fixed point

Not sure if we can make this more "compact".

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  • 3
    $\begingroup$ @China: I don't agree with this answer. My point of view is explained in my answer. $\endgroup$ – Taroccoesbrocco Aug 3 at 10:59
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    $\begingroup$ Well, my answer matches better to your 2. than your 1. So in fact we agree on the negation... $\endgroup$ – zwim Aug 3 at 13:30
  • $\begingroup$ @Taroccoesbrocco: Thank you. "Unique" refers to "rational fixed point" $\endgroup$ – Germany Aug 3 at 14:46
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    $\begingroup$ @zwim - I'm not sure to understand your comment: If $f$ has an irrational fixed point and a rational fixed point, then your version of the negation is true (because of point 3) but my version of the negation (see point 2 in my answer) is false. So, your negation and my negation are not equivalent, and only one of them is correct. $\endgroup$ – Taroccoesbrocco Aug 3 at 15:35
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    $\begingroup$ @China - Then the correct negation is in point $(2)$ of my answer, and not in point $(1)$ (or in zwim's answer). $\endgroup$ – Taroccoesbrocco Aug 3 at 15:45
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The sentence can be interpreted in two ways, depending on whether "unique" refers to "fixed point" or "rational fixed point":

  1. "Unique" refers to "fixed point": The function $f$ has a unique fixed point $a$, and moreover $a$ is rational. The logical form of this sentence is \begin{align} \exists a : a \in \mathbb{Q} \land a = f(a) \land \forall z (z = f(z) \to z = a) \end{align} In this case, the negation is: the function $f$ either has a unique irrational fixed point, or several fixed points, or no fixed points at all (this is equivalent to what @zwim said in his/her answer).

  2. "Unique" refers to "rational fixed point": The function $f$ has exactly one rational fixed point (and possibly many irrational fixed points). The logical form of this sentence is: \begin{align} \exists a : a \in \mathbb{Q} \land a = f(a) \land \forall z ((z \in \mathbb{Q} \land z = f(z)) \to z = a) \end{align} In this case the negation is: the function $f$ has either several rational fixed points or no rational fixed points at all (but it might have irrational fixed points).

Note that the negation of the interpretation $(2)$ is more restrictive than the negation of the interpretation $(1)$: for instance, if $f$ has a irrational fixed point and a rational fixed point, then the negation of $(1)$ is true but the negation of $(2)$ is false.

Unlike @zwim, in my opinion the intended meaning of the sentence should be $(2)$, and not $(1)$, unless contextual information (which is missing in the question of the OP) says differently.

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