# Is “implies” the best symbol when rewriting equations?

In my mathematical homework, I usually indicate algebraic rewrites of equations using implication, and the symbol "$$\implies$$" (LaTeX \implies). For instance, I might write $$3 x - y = 0 \implies 3 x = y \implies x = \frac{y}{3}$$ to mean that, since $$3 x - y = 0$$, the equivalent equation $$3 x = y$$ is also true, which then indicates that $$x = y/3$$ is true. Is logical implication the correct facility to express rewriting an equation into an equivalent form? If not, what other concept and symbol would be correct here?

• If you know the rewrite is equivalent, you can just use $\iff$ (\iff). Implication does not make it visible that the rewrite is reversible. Jul 12, 2021 at 16:42
• Consider $\therefore\,$ (therefore) Jul 12, 2021 at 16:46
• The logical implication (and/or iff) symbol isn't inappropriate, but I sometimes find it a bit "heavy", both visually and cognitively. (After all, the reader is probably sophisticated enough to understand the implications of simple algebraic manipulations.) ... I tend to use "$\to$" (\to) to provide a sense of flow from one version of an equation to another (with generous spacing to help indicate that I'm not constructing some formal logic expression); eg, $$3x-y=0 \qquad\to\qquad 3x=y \qquad\to\qquad x=\frac{y}{3}$$
– Blue
Jul 12, 2021 at 16:56
• @Blue I actually find your comment the most complete answer to my question, due to your mention that "The logical implication [...] symbol isn't inappropriate". If you would care to turn it into an answer, I would be happy to accept it. Jul 12, 2021 at 21:32
• Does this answer your question? Difference between $\implies$ and $\;\therefore\;\;$?
– user1150169
Feb 16 at 0:45

Another symbol which you can use is the "if and only if" symbol, in $$\rm\LaTeX$$, it is $\iff$ and is denoted by $$\iff$$(Also pointed out by @StefanOctavian)

So your equation re-write thus becomes : \begin{align} 3x-y=0 &\iff 3x = y\\ &\iff x = \frac{y}{3} \end{align}

• Note the careful use of spacing and alignment to make this sequence of rewrites easier to to read... that, in my opinion, is as important as the choice of symbol. Jul 12, 2021 at 22:36
• @mweiss Thanks for your advice. I didn't know how to align the statements so I just used the  sign to write block statements but from now on I will be careful about it Jul 13, 2021 at 5:33

1. When solving equations, to emphasise that your work is displaying a chain of equivalent equations, i.e., each step is “reversible”, the correct symbol is $$\iff,$$ not $$\implies$$: $$P(x)=Q(x)\implies x\in\{2,4,5\}$$ merely means that $$2,4,5$$ are candidate solutions, whereas $$P(x)=Q(x)\iff x\in\{2,4,5\}$$ means that the solution set is actually—not merely a subset of—$$\{2,4,5\}.\\$$

2. Outside of the context of writing proofs and specifying theorems, where statements like $$\text{“for each real x,\quad \implies ”}$$ are commonplace, usually the connective that is actually meant is “therefore” rather than “implies”, in which case the correct symbol is $$\,\therefore\;,$$ not $$\implies.$$ (Published work spell “therefore” out because peppering mathematical prose with symbols decreases readability.)

For example, \begin{align}&x=3\\\therefore\; &x\in\mathbb R\end{align} and $$x=3\\x\in\mathbb R$$ are asserting the value of $$x$$ and (explicitly and implicitly, respectively) deducing that $$x$$ is real, whereas \begin{align}&x=3\\\implies &x\in\mathbb R\end{align} is merely specifying that $$x$$ being real is the consequence of a particular case: $$x$$'s value is not being established and $$x$$ possibly actually equals, say, $$2i.$$

If it is a composition of multiple equations it makes sense to say: $$\therefore\tag{therefore}$$ you also have other arrows like: $$\Rightarrow\tag{Rightarrow}$$ which I prefer to use mostly because it is a shorter symbol. As others have mentioned, you also have: $$\iff\tag{iff}$$ $$\to\tag{to}$$ $$\rightarrow\tag{rightarrow}$$ as well as what you have mentioned: $$\implies\tag{implies}$$

It can also sometimes be useful to use symbols like: $$\forall\tag{forall}$$ $$\because\tag{because}$$

EDIT:

One thing I thought I would add because it is especially useful is spacing when writing these equations in one line. Try to use a combination of commands like "," for a short space and "\quad" "\qquad" for longer spaces

• Is there any difference between $\text{(to)}$ and $\text{(rightarrow)}$? Jul 12, 2021 at 20:08
• @A-LevelStudent as far as I am aware, they are the same symbol. Although note that when writing limits the spacing in a subscript seems to work better for \to whilst the spacing in-line works better with \rightarrow Jul 12, 2021 at 22:04
• Alright, thanks. Jul 14, 2021 at 12:52
• If you want a short $\iff$, use $\Leftrightarrow$ (\Leftrightarrow). (I hate the long arrows you get with \implies and \iff.) I also think that $\therefore$ reads far better in examples like the OP's. Jul 15, 2021 at 1:17

The logical implication (and/or iff) symbol isn't inappropriate, but I sometimes find it a bit "heavy", both visually and cognitively. (After all, the reader is probably sophisticated enough to understand the implications of simple algebraic manipulations.)

I tend to use "$$\to$$" (\to) to provide a sense of flow from one version of an equation to another (with generous spacing (\quad or \qquad) to help indicate that I'm not constructing some formal logic expression); eg, $$3x-y=0\qquad\to\qquad 3x=y \qquad\to\qquad x = \frac{y}{3}$$

The traditional symbol for "therefore" is $$\therefore$$ (\therefore in $$\LaTeX$$). I recommend it in your example, because "$$A$$ implies $$B$$" and "$$A$$ therefore $$B$$" don't mean the same thing: the former means that if $$A$$ holds, then so does $$B$$, the latter means that $$A$$ holds, and as a consquence of that so does $$B$$. In your example, with $$A \equiv 3x - y = 0$$, $$B \equiv 3x = y$$ and so on, it is the latter reading that you want. For a simpler example: "$$0 = 1 \implies 1 = 1$$" is true, but "$$0 = 1 \therefore 1 = 1$$" is false.

(Technically $$\therefore$$ is just logical conjunction, but with an implied hint that the right conjunct is easily derivable from the left conjunct.)