How do I know if this statement is true or false?
How do you read something like that:
$$\forall x\exists y\mid x^{2} = y$$
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1For every $x$ there exists $y$ such that $x^2=y$ – kingW3 Sep 13 '19 at 00:18
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There should be domains for $x$ and $y$.If it was $\forall x \in \mathbb{N},\exists y \in \mathbb{N}: x^2 = y $, it would be true and will read "for all x in $\mathbb{N}$", there exists $y$ in $\mathbb{N}$ such that $x^2 = y $.
underdisplayname
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You are right. But here, I'm saying that there exists $y$ s.t. it is the square of the natural number. – underdisplayname Sep 13 '19 at 02:44
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It is false if $$ \lnot (\forall x \exists y \mid x^2 = y) $$ or, equivalently, $$ \exists x \forall y \mid x^2 \neq y $$ is true. To prove this statement, you must find some $x$ such that $x^2 \neq y$ for any $y$.
A proof that the statement is true might look something like this: Fix an arbitrary $x$. Find a $y$ (which could depend on $x$) susch that $x^2 = y$. Since $x$ was arbitrary, we may conclude that the statement is true.
Charles Hudgins
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