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I've used both $\Rightarrow$ and $\implies$ interchangeably throughout my mathematics in school, and I want to know which is proper. When should I use $\Rightarrow$ over the implies arrow? Does it mean leads to? Say I have $x+3=4$. Would I say:

$x+3=4\Rightarrow x=1$

or

$x+3=4\implies x=1$

Since the equation doesn't really 'imply', it leads to the solution that $x=1$. I haven't been able to find anything on this online, so I would hope it could be cleared up here :)

jackson
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  • The statement $x+3 = 4$ does imply the statement $x=1$. Use $\implies$ in most cases. – Qi Zhu Jun 22 '22 at 05:12
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    While "... and therefore ...." and ".... implies ..." maybe be subtly different and while there may be typesetting differences withing a text between the two different arrows for each, there is simply no way there is universe standards for typesetting over different media. It is not reasonable to expect anyone to even see the difference between $\Rightarrow$ and $\implies$ much less to know which one the author intended for which purpose. – fleablood Jun 22 '22 at 06:16
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    no significant differences in meaning due only to the long and short form of that symbol. – Mauro ALLEGRANZA Jun 22 '22 at 06:31

2 Answers2

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$x+3=4\implies x=1$ Since the equation doesn't really 'imply', it leads to the solution that $x=1$.

Take any $x.$ Satisfying $$x+3=4$$ implies that it also satisfies $$x=1.$$

That is, the solution set of the first equation is a (possibly proper) subset of the solution set of the second equation.

(So, $x^2=25\kern.6em\not\kern-.6em\implies x=5.)$

I've used both $\Rightarrow$ and $\implies$ interchangeably throughout my mathematics in school, and I want to know which is proper.

Did you mean to contrast $\Rightarrow$ and $\rightarrow$ instead?

$\Rightarrow$ and $\implies$ mean exactly the same: they are just different “handwriting”.

ryang
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    Nope, I meant the short and long. Yeah, just different handwriting thanks!!! For $\rightarrow$ vs $\Rightarrow$, I'm a bit unclear on the difference, so if you could specify that would be great :) – jackson Jun 22 '22 at 05:34
  • By $\to$ and $\longrightarrow,$ I don't think you mean the material conditional or mapping of sets, in which case it's probably just diagrammatic, highly informal notation: typically not necessary within the text, but useful in the scribblings around the text. – ryang Jun 22 '22 at 05:45
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    In my mind it'd be totally unreasonable to expect anyone to see a difference between $\Rightarrow$ and $\implies$ much less to expect there to be a universally agreed upon meaning for either. – fleablood Jun 22 '22 at 06:18
  • Could you explain why $x^2=25\not{\implies} x=5$? Also how did you type the not implies symbol, the formatting on mine is clearly wrong, using \not\implies. – jackson Jun 24 '22 at 03:33
  • @jackson 1. Here's an explanation that I wrote. In brief: if $x=-5,$ and you are privy only to the fact that $x^2=25,$ then you'd be wrong to be confident that $x=5,$ and in fact, the correct deduction, premised on your incomplete information, is that $x$ must be either $5$ or $(-5).\quad$ 2. $\kern.6em\not\kern-.6em\implies$ gives $\kern.6em\not\kern-.6em\implies.$ – ryang Jun 24 '22 at 05:12
  • @jackson 3. "Is there a preferred one of the two \Longrightarrow and \implies due to the spacing? When would I use one and not the other?" Both of our answers are pointing out to you that there is neither semantical nor syntactical difference between these symbols. My white dog being named Brownie is just a quirk, and its meaning is perhaps sentimental, but neither semantical nor physiological. – ryang Jun 24 '22 at 05:32
  • @jackson 4. Many symbols in LaTeX have multiple codenames (e.g., \to \longrightarrow) that produce EXACTLY the same result; I never use x\Longrightarrow y and instead always just type x{\implies}y when I don't want the excess space around that symbol to achieve EXACTLY the same effect. $\quad$ 5. Many symbols in LaTeX have multiple versions, and making a choice is frequently a matter of readability, consistency and style. The difference in this case is analogous to the difference between ( and [; I will variously type $(;\big(;\Big(;\bigg($ depending on my needs. – ryang Jun 24 '22 at 05:34
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There are 3 different levels to consider:

1. Mathematical semantics: Both arrows mean "implies" or "from $A$ follows $B$". If you'd use the symbols with paper+pencil or blackboard+chalk, they are merely indistinguishable, because noone would start measuring their length/height ratio.

2. Typography: Here is the main difference. The short $\Rightarrow$ fits better with inline text, wheras the longer $\implies$ goes better with separated formula and additional spacings like $$\text{it's raining} \quad\implies\quad \text{the streets are getting wet}$$

There are actually 3 variants of this arrow:

  • $x\Rightarrow y\qquad$ x\Rightarrow y
  • $x\Longrightarrow y\qquad$ y\Longrightarrow y
  • $x\implies y\qquad$ x\implies y

As you can see, \Longrightarrow has less spacing around it than \implies.

3. Semantics of the LaTeX source: Glyphs named \implies or \iff can add semantics to the source code. However, source semantics is far from being perfect, so that this point is only mentioned for completeness.

emacs drives me nuts
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  • Thanks! Is there a preferred one of the two \Longrightarrow and \implies due to the spacing? When would I use one and not the other? – jackson Jun 24 '22 at 03:37