truth value of the following statement: $\exists x\forall y\neq0(xy=1)$

Question

I'm trying to figure out the truth value of the statement, and the correct way to read it. I'm reading it as :" there exists $x$, such that for all values of $y$ not equal to $0$, $xy=1$". So I think its false. Can someone please provide feedback

Answer 1

The symbols $\exists$ and $\forall$ need to have a domain, and presumably it is implicit in the problem you are looking at. If this domain is nice then your interpretation is correct. For example, $$\exists x\in\mathbb{R}\:\forall y\in\mathbb{R}\:(xy=1)$$ is a false statement (as $x$ is fixed so there is at most one $y\in\mathbb{R}$ such that $xy=1$, namely $1/x$).

On the other hand, $$\exists x\in\{1\}\:\forall y\in\{1\}\setminus\{0\}\:(xy=1)$$ is a true statement.

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Finally, note that swapping the quantifiers produces a true statement (assuming a nice domain): $$\forall x\in\mathbb{R}\setminus\{0\}\:\exists y\in\mathbb{R}\:(xy=1)$$