Alternate translation for: “Every real number except zero has a multiplicative inverse.”

Question

A given text states, “Every real number except zero has a multiplicative inverse" (where mul- tiplicative inverse of a real number x is a real number y such that xy = 1).

It offers the following translation:

$$\forall x((x\neq 0) \rightarrow \exists y(xy = 1)).$$

I personally translated the statement as:

$$\forall x \exists y((x\neq 0)\rightarrow (xy = 1)).$$

Are these two statements logically equivalent? My reasoning being, for every real number x, there exists a real number y, such that if x does not equal zero, then the product of x and y equals 1.

Answer 1

Yes :

$∀x((x \ne 0 ) → ∃y(xy = 1 ))$

and

$∀x∃y((x \ne 0) → (xy = 1))$

are logically equivalent, because :

$\vdash \exists y (\alpha \rightarrow \beta) \leftrightarrow (\alpha \rightarrow \exists y \beta) \quad $ if $y$ is not free in $\alpha$.

In your case, $\alpha$ is $(x \ne 0 )$ and $y$ is not free in it.

Answer 2

Yes, it is a general principle that if $y$ does not appear in $\varphi$, then the following are equivalent.

1. $\varphi \rightarrow \exists y(\psi)$
2. $\exists y(\varphi \rightarrow \psi)$

Answer 3

Yes, it is correct. This principle is known as "null quantification rule". In this case, the left-hand $x$ is independent of domain $y$. So, we can place $y$ domain on left hand side. Follow the below link to get more details about null quantification rule. https://gateoverflow.in/130504/null-qunatification-rule