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I need to understand the meaning of the FOL statement below.

$$ \forall x \exists y \forall z (z \neq y \iff f(x) \neq z) $$

Does this imply that $x$, $y$, and $z$ cannot be the same or $f(x)$ has no value?

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The statement is

"for all $x$, there exists a value of $y$ such that for all $z$,
$z\neq y$ if and only if $z \neq f(x)$".

This can be simplified: $$\begin{align} & & \forall x \exists y \forall z (z\neq y \iff z \neq f(x))\\ &\implies & \forall x \exists y \forall z (z=y \iff z = f(x))\\ &\implies & \forall x \exists y \forall z (y = f(x))\\ &\implies & \forall x \exists y (y = f(x))\\ \end{align}$$

If we denote the set of all values of $x$ by $X$ and the set of all values of $y$ by $Y$, then this tells us that the function $f$ maps every $x$ in $X$ to a $y$ in $Y$. That is, $f: X \to Y$.

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  • $\begingroup$ Thanks a lot Philip, I got that for each value of x there is a corresponding y mapping, will there be a case such that- 1. x,y,z cannot be same, 2. for each argument x, f(x) is unique, 3) y indicates that f(x) has no value .. $\endgroup$
    – ammu
    Apr 20, 2019 at 4:54
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    $\begingroup$ No to all. The first two are more restrictive than the given statement. For 1, let f be the identity mapping. For 2, f could be a constant function. I'm assuming all variables share the same domain. 3 is in direct contradiction with the statement, which says that f(x) always has a value. $\endgroup$ Apr 20, 2019 at 5:08
  • $\begingroup$ @ammu Don't forget to accept answers by clicking the check mark if they solved your problem. $\endgroup$ Apr 20, 2019 at 18:45

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