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Consider the statement "Every real number except zero has multiplicative inverse."

  1. My Discrete Mathematics textbook translates this statement to $$\forall{x}{({(x\neq0)} \boldsymbol\rightarrow{\exists{y}{(xy=1)}})}.\tag1$$ I think that the above translation is correct if and only if you interpret the given statement as not specifying whether zero has multiplicative inverse, that is, that all numbers have inverses while zero may or may not have an inverse; in this vein, "Every real number greater than 7 has a multiplicative inverse" translates to $\forall{x}{({(x>7)} \rightarrow{\exists{y}{(xy=1)}})}$ and doesn't mean that only real numbers greater than $7$ have a multiplicative inverse.

  2. However, if you interpret the given statement as meaning that every real number except zero has multiplicative inverse while zero does not have a multiplicative inverse, then this translates to $$\forall{x}{({(x\neq0)} \boldsymbol\leftrightarrow{\exists{y}{(xy=1)}})}.\tag2$$

My professor insists that the book's translation is correct, explaining that we know that zero doesn't have a multiplicative inverse.

Is my above argument correct, or is the first translation indeed the only correct one, or is the given statement actually ambiguous?

ryang
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Walid Amro
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    You are correct. – Brian M. Scott Jul 01 '20 at 17:19
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    It is possible to say, that you both are right, because both sentences are correct. If we know, that equivalence is correct, then implication is correct also. Your one implies other, of course. – zkutch Jul 01 '20 at 17:22
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    Usually when you publish a theorem like $X \implies Y$, you want the weakest $X$ and the strongest $Y$ you can. Because $A \implies B$ is weaker than $A \iff B$, it is preferred as an assumption, as long as it actually works. – DanielV Jul 01 '20 at 20:23
  • @DanielV Could you please expand on that a little or point me in the right direction to read more on the subject. Thank you. – Walid Amro Jul 01 '20 at 22:26
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    @PsychicLegend What I wrote is so mundane that I doubt anyone has written much on it. It is the basis of Hoare logic but that is way beyond the scope of your question. Is there some aspect of what I said that wasn't clear to you? Like is the concept of strong/weak propositions clear? – DanielV Jul 02 '20 at 00:17
  • @DanielV Yes. I know that implies is weaker than iff but I didn't completely understand what you meant by the weakest $X$ and the strongest $Y$. Now i know (correct if I'm wrong here) that for example p="I am student" is weaker than q="I am a student and I love basketball". If we took these two examples and applied what you said on them then we get $p\rightarrow q$ which, the way I see it, is wrong and it must be $q\rightarrow p$ since if I was in a more specific case (stronger proposition) then I am for sure in the more general one (weaker proposition) but not vice versa. – Walid Amro Jul 02 '20 at 08:09
  • @DanielV So, my questions are:
    1. Is my understanding of strong and weak proposition correct
    2. and if so, wouldn't we want strong implies weak instead of the opposite

    Thank you for being patient and sorry if I'm asking a lot of questions but I always like to understand the topic at hand the best I could.

     – Walid Amro Jul 02 '20 at 08:09
  • @DanielV Just one more question. I get that one must use the weaker form a compound proposition if he's not sure of the correctness of the stronger one, but why would we want to use the weaker form of a compound proposition if we know for a fact that the stronger form is more correct and more true to real life. – Walid Amro Jul 02 '20 at 08:19

2 Answers2

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If you can establish that $\forall x . x \cdot 0 = 0$, then that already implies $\forall x \bigg( x \ne 0 \leftarrow \bigg(\exists y. xy=1\bigg) \bigg) $ . So the $\leftarrow$ direction of the $\iff$ is usually already redundant. But it isn't wrong, so all that is left is style.

And as a general point of style and usefulness, search for assumptions that are as weak as possible but still strong enough to be sufficient. Aim for conclusions that are as strong as possible, but weak enough to still be correct.

The set of natural numbers that is divisible by 6 is a strict subset of those that are divisible by 2. So $6|x$ is a strictly stronger claim than $2|x$. If I tell you $6|x$ then I have told you "more information" than if I have only told you $2|x$. On the other hand, if I say "I will let you out of jail if you find me an $x$ divisible by 6", then I am asking more of you than the other jailer who would release you for only finding and even $x$.

Strong/weak is a strictly partial order. For the two statements $x > 1000$ as well as $x \text{ is even}$, neither is stronger or weaker than the other. The set of numbers satisfying the first is neither a superset nor a subset of the second.

If you are offering the theorem $X \to Y$ to the world: If $X$ is too strong, no one can ever use it. If $Y$ is too weak, there is no reason to use it. An example of a very useless theorem would be $z = 178462827 \implies z = z$. Who cares about $178462827$? That is too strong of a requirement. And $z=z$ is such a weak claim that is always true. Not helpful.

Implications $X \to Y$ are 3 parts: the assumption $X$, the conclusion $Y$, and the implication $X \to Y$. When you offer a theorem, you want to offer the strongest not-wrong version of the theorem. An implication is strengthened by weakening the assumptions. An implication is also strengthened by strengthening the conclusions.

The weakest possible implication is $\text{false implies true}$. It is so weak that it manifests both meanings of "vacuously true".

Now as a disclaimer, there is some cheating going on here, because every mathematical theorem when written out with all of the assumptions that goes into it is a tautology, so in that context every mathematical theorem is technically equally strong. So if you wanted to get into formal logic, all this only applies to domain specific way theorems are presented, not to the entire tautological version of theorems.

Now i know (correct if I'm wrong here) that for example p="I am student" is weaker than q="I am a student and I love basketball".

Correct.

DanielV
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  1. The word "except" flags members within a reference set for which the predicate (property) is false. It does not omit these members from predication.

    Thus,

    • Every number except zero has a multiplicative inverse

    means $$\forall{x}{({(x\neq0)} \leftrightarrow{\exists{y}{(xy=1)}})}.\tag2$$

  2. In contrast,

    • Every non-zero number has a multiplicative inverse

    means $$\forall{x}{({(x\neq0)} \rightarrow{\exists{y}{(xy=1)}})}\tag1$$ and is a weaker assertion that doesn't actually specify whether zero has a multiplicative inverse.

Your professor's argument is anyway invalid: a translation exercise is about being faithful, not about being factual.

ryang
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