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Let $P(x,y)$ be the proposition $x^2=y$, where $x$ and $y$ are integers. Determine the truth value of each of the following proposition.

for $∃x P(6,x)$

Does this mean that 6 is $x$ in this case and $x$ is $y$ So $y$ would equal to $6^2=36$ How do I know if that's true?

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  • $\begingroup$ What is the "for" there for? The proposition $\exists x P(6,x)$ is meaningful (and true; take $x=36$.) $\endgroup$ – Mark Fischler Aug 29 '17 at 21:42
  • $\begingroup$ There may or may not have been a typo $\exists x, P(6,x)$ means "There is an integer $x$ such that $6^2 = x$". $\exists x, P(x, 6)$ means "There is an integer $x$ such that $x^2 = 6$". $\endgroup$ – fleablood Aug 29 '17 at 21:47
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You are exactly correct. The names of the variables do not matter. The position of the variables is the important thing.

So $\exists{x}P(6,x)$ is equivalent to $\exists{y}P(6,Y)$

and also to $\exists{t}P(6,t)$ and so on.

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  • $\begingroup$ It may be helpful to compare $\exists{r}P(6,r)$ which is True with $\exists{v}P(v,2)$ which is false in the domain of discourse: the integers. $\endgroup$ – Jim H Aug 29 '17 at 21:52

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