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I always have this problem where I want to write an equation then immediately its simplification after it. I used to use the imply or equivalent symbol such as:

Therefore we end up with $$ \frac{a^2}{c^2} + b^2 =1 \implies a^2 + b^2c^2 = c^2. $$

But I was told this was not good in formal settings. Instead I should do something like:

Therefore we end up with $$ \frac{a^2}{c^2} + b^2 =1, $$ and hence $a^2 + b^2 c^2 = c^2$.

I understand why this would be better. Sometimes I can skip the penultimate equation and that would be fine. But other times I do find it a lot more clear for the reader to see that extra step. And yet the words in between the equations seems very distracting and not needed. Is there a symbol that I can put between the equations that doesn't carry a lot of meaning and that is also formal? Or is this just a bad idea altogether and I should always connect the two equations with words?

Edit: BTW I have looked at this question but it didn't help. I would never write a proof with imply symbols. This is purely about a step between two equations that is trivial enough not to require explanation yet the text would be less clear without it.

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    $\begingroup$ The relevant statement in your example is the end result $\,a^2+b^2c^2 = c^2\,$, rather than the logical implication $\,\frac{a^2}{c^2} + b^2 = 1 \implies a^2 + b^2c^2 = c^2\,$. Writing it in words ("hence") makes this more clear than using symbols ("$\implies$"). You could use the therefore sign, instead, $\,\frac{a^2}{c^2} + b^2 = 1 \quad\therefore\; a^2 + b^2c^2 = c^2\,$, though that's a matter of personal style and opinion. $\endgroup$
    – dxiv
    Feb 21, 2022 at 1:58

1 Answer 1

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Outside of the context of specifying theorems, where statements like $$\text{“for each real $x,\quad$ <conditions> $\implies$ <conclusion>”}$$ are commonplace, usually the connective that is actually meant is “hence/therefore/thus” rather than “implies”, in which case the correct symbol is $\,\therefore\;$ instead of $\implies.$

(Published work spell “therefore” out because peppering mathematical prose with symbols decreases readability.)

For example,

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1,$$ and hence $$a^2 + b^2 c^2 = c^2.$$

or something like

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1;\\\therefore a^2 + b^2 c^2 = c^2.$$

or even

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1\\ a^2 + b^2 c^2 = c^2.$$

each—as intended—asserts both equalities, whereas

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1 \implies a^2 + b^2c^2 = c^2.$$

merely asserts that $a^2 + b^2c^2 = c^2$ is a consequence of $\frac{a^2}{c^2} + b^2 =1,$ while stating neither the former nor the latter to be true. This last presentation is momentarily confusing—and at least jarring— because I do literally read it as “Therefore we end up with P implying (i.e., being a sufficient condition for) Q.”


Addendum

Page 17 of this style guide directly addresses your question:

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    $\begingroup$ I really don't like the other two examples. I've never seen the triple dot used in a written piece of work (it looks rather ugly) though it may be acceptable for blackboard lecturing. Not using any connecting symbol looks even worse as the relationship with the two lines is not even clear. With using the $\implies$ symbol, I agree it's not ideal (it's better to write the words imo), but the meaning is quite clear from context, to me at least. $\endgroup$
    – YiFan Tey
    Feb 21, 2022 at 5:21
  • $\begingroup$ I also think often $\implies$ is used to mean "which implies", and that makes the usage of the symbol much more acceptable. $\endgroup$
    – YiFan Tey
    Feb 21, 2022 at 5:22
  • $\begingroup$ Yes yes triple dots is ugly (hence my parenthetical preface), and yes without connectives is arguably a tad ambiguous (but not much), and yes $P{\Rightarrow}Q$ can alternatively be read as "$P,$ which implies $Q$". However, this semantically conflicts with the standard "$P$ implies $Q$", so whenever an author intends this secondary interpretation, I always end up having to re-parse the statement, which breaks the flow. $\endgroup$
    – ryang
    Feb 21, 2022 at 10:47
  • $\begingroup$ Yes, I agree it is better to write it out in full using words so that it doesn't break the flow or lead to problematic parsing. $\endgroup$
    – YiFan Tey
    Feb 21, 2022 at 12:26
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    $\begingroup$ What is your opinion on putting the words inside the display math environment like: $$ \frac{a^2}{c^2} + b^2 = 1, \quad \text{and hence}\quad a^2 + b^2 c^2 =c^2. $$ $\endgroup$
    – Jean
    Feb 21, 2022 at 15:22

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