What is the negation of this sentence

Question

What is the negation of the following case:

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

Answer 1

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".

Answer 2

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.