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I want to find the truth value of this statement given that the domain of discourse is $\mathbb Z$:

$$∃m∃n (2x+5y=y ∧ x+5y=3).$$

I am a bit confused whether the statement is correct or not since $x$ and $y$ are not quantified (we did not say $∃x$ or $∀x$). If it is correct not to quantify them, then we can deduce that the statement is true for $x=-2$ and $y=1.$ But I am unsure if this is the right way to solve this problem.

ryang
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joumana
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    If the variables m and n do not occur in the formula, then the quantifiers ∃m ∃n are useless. – Mauro ALLEGRANZA Sep 27 '23 at 09:52
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    Maybe you have a system of two equations: 2x+5y=y and x+5y=3 and you have to solve it... – Mauro ALLEGRANZA Sep 27 '23 at 09:53
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    It is correct to say that that quoted line has no definite truth value and is true for $(x,y)=(-2,1).$ Anyway, the m and n are almost certainly typos (and intended as x and y), in which case exhibiting $(x,y)=(-2,1)$ proves that the quoted statement is true. – ryang Sep 27 '23 at 13:16

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