2

Can the English sentence, "every number other than zero has a multiplicative inverse" be written as $∀x(x \ne 0 → x \cdot 1/x = 1)$

cpiegore
  • 1,556
Kheaven
  • 21
  • 1
  • 3
    x factorial is never zero. Use $\ne$ for the not-equals operator. != is a programming language thing. – Dan Jul 09 '22 at 21:16
  • 1
    For the statement in the title, I would say that you are still using two "variables": $x$ and $1/x$. – Sassatelli Giulio Jul 09 '22 at 21:25
  • For the statement in the question, the short answer is no. – Sassatelli Giulio Jul 09 '22 at 21:27
  • $1/x$ multiplied by $x$ equals one from the definition of $1/x$. If you can write $1/x$ then you don't need to check if this multiplication is equal to one. – Mateo Jul 09 '22 at 21:45
  • The english sentence has a distinct subject "every number other than $0$" so that requires a variable, and it has a distinct object "has a multiplicative" inverse. That requires second variable. Now you can notate the variable in terms of the first such as $x$ and $y_x$ or $x$ and $x^{-1}$ or even $x$ or $\frac 1x$. However I'd advise against that latter two as they come with "emotional baggage" but even if you do you MUST consider $\frac 1x$ a second and different variable than $x$. (so if you did the you'd say $\forall x(x\ne0\to\exists\frac1x:x\cdot \frac 1x=1)$. (But I wouldn't) – fleablood Jul 10 '22 at 01:01

2 Answers2

3

I would denote the statement as:

$$\forall x (x\neq 0\implies\exists y \text{ such that } y\cdot x = x\cdot y = 1)$$

As Mateo already said, you cannot write $1/x$ before you show that it even exists. In the case of my notation, you can conclude $y=1/x$.

Andreas Tsevas
  • 2,639
  • 6
  • 15
0

What does $1/x$ mean? In order to write your sentence, you assume the existence of a function $f(x)=1/x$, which you assert has a certain property ($x \cdot 1/x =1$). This is different from the sentence you want to write, which says that for any nonzero number, there is another number such that the two numbers multiply to 1.

Alex Mine
  • 188