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Let's say we have the expression

$$∀,∈ℤ:=⟹=$$

Which means "for all values of and in the set of integers, if equals then must equal ."

For example, if 4×6=4×6, then 4=4.

However, this does not apply to 0.

For example, 7×0=6×0, but 6≠0.

Is there a way to add to the previous expression a term that says "as long as x doesn't equal 0"?

Alex
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The_Animator
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    Yes, then we write $\forall a,b\in \Bbb Z$ and for all $x\neq 0$ we have that $ax=bx$ implies $a=b$ by cancelling. – Dietrich Burde May 31 '23 at 16:13
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    There are a few ways to do this; in your example, you could instead say $\forall a,b\in\mathbb{Z}, ax=bx\implies (a=b\lor x=0)$ (where $\lor$ is read as "or"), or equivalently $\forall a,b\in\mathbb{Z}, ax=bx\land x\neq0\implies a=b$ (where $\land$ is read as "and"). – Charlie May 31 '23 at 16:15
  • "a term that says "as long as a or b doesn't equal 0"? Be careful, you want to cancel $x$, which should be nonzero. You don't have to assume anything else on $a$ and $b$. – Dietrich Burde May 31 '23 at 16:20
  • @DietrichBurde Ah, you're right. Sorry, I meant to say as long as x doesn't equal 0. Thank you for pointing that out! – The_Animator May 31 '23 at 16:22
  • Then the answer is easy - just add another $\forall x\neq 0$ in front. And say, what kind of number $x$ should be, e.g., $x\in \Bbb Z$, or $x\in \Bbb F_p(T)$. – Dietrich Burde May 31 '23 at 16:23
  • Or: $∀{,}{∈}ℤ;:∀x{∈}\mathbb R{\setminus}{0};;(=⟹=).$ The colon is unnecessary. – ryang May 31 '23 at 16:23
  • @ryang Yes, this is what I wanted to say! Only perhaps $x\in \Bbb C\setminus 0$ instead, or whatever domain we are considering. – Dietrich Burde May 31 '23 at 16:24
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    @ryang Ah, I see! Thank you for clearing this up! :D – The_Animator May 31 '23 at 16:26
  • Now that I think about it, "∀,∈ℤ:≠0⟹(=⟹=)" works. – The_Animator Sep 25 '23 at 17:27

2 Answers2

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There is no formal symbol for this. Even if there were, words are usually easier on your reader than formality. So write

If $x \ne 0$ then $ax= bx \implies a =b $.

You could write

$ax= bx \implies a =b $ provided $ x \ne 0$.

but that might make your reader hesitate and wonder until they finished the sentence. It's kinder to state the condition first.

The Ruby programming language has an unless keyword that can follow the statement it references, so you could code something like

you can cancel x unless x is 0
Ethan Bolker
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  • $\forall { a, b, x } \in \mathbb{R} \wedge x \neq 0, a x = b x \implies a = b.$ – David G. Stork May 31 '23 at 16:44
  • @DavidG.Stork That's correct, of course, but much harder to read than the same thing written out mostly in English. – Ethan Bolker May 31 '23 at 16:46
  • Or: $∀{,}{,}x{∈}\mathbb R;(x\ne0\land =⟹=).$ @DavidG.Stork – ryang May 31 '23 at 16:55
  • Btw, do "as long as" and "provided that" both simply mean "if", or are there exceptions? Some online dictionaries claim that "as long as" mean "only if", but then again they anyway seem to variously think that "only if" means "if" or sometimes "iff". – ryang May 31 '23 at 16:57
  • @ryang Yes, but see my response to David Stork. This is all part of my campaign to convince students that "prove it" does not mean "write it using formal logic". – Ethan Bolker May 31 '23 at 16:58
  • @ryang Those are both perfectly good synonyms for "if". Choose what reads best in any particular argument. – Ethan Bolker May 31 '23 at 16:59
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You could use "as long as $x$ does not equal $0$" like so:

With $\forall x \neq 0$ at the end:

$∀a,b\in\mathbb{Z}:ax=bx\implies a=b, \forall x \neq 0$

or with $\forall x\in\mathbb{Z}^*$ at the end:

$∀a,b\in\mathbb{Z}:ax=bx\implies a=b, \forall x\in\mathbb{Z}^*$

or with $\forall x\in\mathbb{Z}^*$ at the beggining:

$∀a,b\in\mathbb{Z} \wedge \forall x\in\mathbb{Z}^*:ax=bx\implies a=b$

Antonio De Angelis
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