How do I put this into an expression "For any nonzero real number $x$, there exist $-x$ real number."?

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How do I put this into an expression "For any nonzero real number $x$, there exist $-x$ real number."? Is it "$\forall x\ne0 \exists(-x)$"?

I don't know how to put "real number $x$" into an equation

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1 Answer

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It looks like what you are trying to do is to formulate the statement that every number $x$ has an additive inverse, denoted $-x$.

A better way to do this is to separate the statement from the definition of the notation, like this:

$\forall x \in \mathbb R \,\,\exists y \in \mathbb R$ such that $x+y=0$. Denote $-x=y$.

However, this raises a logical issue: how do we know that $-x$ is well-defined?

To answer that, one must also state (and prove) uniqueness:

Only after that is done is it logically valid to introduce a well-defined notation for the additive inverse.

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