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I have the following proposition in the domain of nonnegative integers:

$$ \forall x \exists y \cdot(2x-y = 0) $$

I translate it as the following: For all $x \in \{0, 1, 2,3,.... \}$, there is always a $y$ to be found, such that $y=2x$. The answer given is True, but I do not understand how this will be true for the case when $x=0$.

Am I correct in assuming that in predicate statements like this, $x$ and $y$ have to be different?

senseiwu
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    You are incorrect. We can have $y=x$ without any problem. – Taladris Feb 15 '22 at 08:41
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    "Am I correct in assuming that in propositional statements like this, x and y have to be different?" NO. Thus, for $x=0$ we have that $y=0$ satisfies the equation $2x-y=0$. – Mauro ALLEGRANZA Feb 15 '22 at 08:42
  • Thanks, saved my day! – senseiwu Feb 15 '22 at 08:43
  • Ill check if I can find a reference to write a neat self answer – senseiwu Feb 15 '22 at 15:27
  • No kidding, i went through several lectures and also a whole chapter on this topic, nowhere it was mentioned. It made me think that x and y have to be different if quantifiers are there – senseiwu Feb 15 '22 at 17:14
  • @senseiwu Food for thought: in the sentence $∀x P(x){\implies}∀x Q(x),$ is there 1 variable, 2 distinct variables that cannot refer to the same object, or 2 variables that can refer to the same object? P.S. Type @user to notify them of comments – ryang Feb 15 '22 at 20:15
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    @ryang In a strict sense, I see only 1 variable. But they can refer to the same object in the domain of discourse. Please correct me where my thinking is flawed! – senseiwu Feb 15 '22 at 21:39
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    @senseiwu In an even stricter sense, many logic systems consider $∀x P(x){\implies}∀x Q(x)$ to be malformed, and will insist that it be rewritten as $∀x P(x){\implies}∀y Q(y);$ and yes, as in your main question, these two variables indeed can refer to the same object. – ryang Feb 16 '22 at 04:01
  • @ryang Found this which I believe makes my question a duplicate. The statement in the answer "For every " means for every , period. So, yes, there are no restrictions on not being equal to ; any such restrictions would have to be given as predicates (a clause ≠). answers my question. The order of quantifiers are reversed there, but I think the idea is the same. So delete or keep mine? – senseiwu Feb 16 '22 at 07:25

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