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I often wonder about notation and what is acceptable. I have seen many different ways of linking equations together, sometimes with just $=$ and others with $\iff$ or $\implies$. Now, I know when solving a limit or a derivative for example, you can link using only equals signs.

$$ \frac{d}{dx}\left[\frac{x^2}{3x}\right] = \frac{2x(3x) - 3(x^2)}{9x^2} = \frac{6x^2 - 3x^2}{9x^2} = \frac{x^2}{3x^2}=\frac{1}{3} $$

However, when solving a two sided equation then it becomes ugly to use $=$, so I stick to $\implies$ where I have seen others use $\iff$.

$$ \sin x = 1 - \cos^2 x \implies \sin x - (1 - \cos^2 x) = 0 \implies \sin x - \sin^2 x = 0 \implies \sin x(1 - \sin x) = 0 \implies x = n\pi \text{ or } x = 2\pi n + \frac{\pi}{2}$$

Some others seem to think this is abuse of $\implies$, and it should be reserved only for cases such as

$$ \log_b x = y \iff b^y=x $$

When should I use $\implies$ and $\iff$ where it will be accepted by the majority?

Liam McInroy
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    First of all $\lim_{x\to 0^+}\implies x>0$ is nonsense. Even if the limit was completed on the left side, the $x$ in the expression is a bound variable, so it can't imply anything about $x$. – Thomas Andrews Oct 22 '14 at 01:50
  • Technically, the only thing wrong with using $\implies$ here is that you haven't shown $x=n\pi$ or $x=2n\pi + \frac{\pi}2$ are solutions, only that those are the only candidates for solutions. In this case, you could easily use $\iff$ in each step... – Thomas Andrews Oct 22 '14 at 01:54
  • @ThomasAndrews With the limit, that was only when working with limits that approach infinity as justification that it was positive infinity. And does the lack of $\iff$ actually imply that those were not the only solutions? – Liam McInroy Oct 22 '14 at 01:58
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    There is no implication there with the limit. $x$ is not a quantity that makes any sense to talk about when we say $\lim_{x\to 0+} f(x)$. It just doesn't make any sense to conclude anything about $x$, since $x$ doesn't exist. It's like saying $\sum_{i=1}^n a_i = 12\implies i>0$. That doesn't mean anything. – Thomas Andrews Oct 22 '14 at 02:00
  • @Thomas Huh... And why does $\sin x (1 - \sin x) = 0 \iff x = n\pi \text{ or } x = 2\pi n + \frac{\pi}{2}$ correct but not only the $\implies$ ? – Liam McInroy Oct 22 '14 at 02:03
  • Single implies is true. If if $f(x)=0\implies x=0\text{ or } x=1$, that does not mean that $x=1\implies f(x)=0$ or $s=0\implies f(x)=0$. So you haven't necessarily found the solutions if you only know $f(x)=0\implies x=0\text{ or } x=1$. – Thomas Andrews Oct 22 '14 at 02:09
  • In other words, all of you deductions are true above, but $\implies$ means that the far right hand side might contain some non-solutions. – Thomas Andrews Oct 22 '14 at 02:10
  • @ThomasAndrews Ah, so this only builds on the point that Chantry made about how precision – Liam McInroy Oct 22 '14 at 02:12
  • Is “implies” the best symbol when rewriting equations? – ryang Jan 03 '22 at 09:30

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I have made the mistake of using $\implies$ instead of the biconditional in the past.

Essentially, if you are trying to show equivalences, then it is a good idea to use the biconditional. Otherwise, users may become confused, because they will interpret it as meaning that the former implies the latter, but not the other way around.

For example, $x = 0 \implies \sin{x} = 0$ makes sense because there are other solutions for $x$. In other words, $\sin{x} = 0 \implies x = 0$ is false.

But if you were to state $x - 3 = 2 \implies x = 5$, you might not be choosing the best formatting, because it's also true that $x = 5 \implies x - 3 = 2$. The statement is not wrong, but it is not as strong as it could be. It leaves it open whether or not $x - 3 = 2$ when $x = 5$.

I think you already knew this. Maybe it is a little anal pointing this out, but if you are going to write something, why not write it properly?

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    Though one could say you should write what you need and no more. If you only need the forward implication, why write the reverse implication? On the other hand, often you would need the reverse implication, so writing only the forward implication would not actually show what you want to. – Joel Bosveld Oct 22 '14 at 01:55
  • @JoelBosveld Very true. –  Oct 22 '14 at 01:57
  • I really like the point both you and Thomas Andrews made about precision, +1 – Liam McInroy Oct 22 '14 at 02:12