Is “implies” the best symbol when rewriting equations?

Question

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?

Answer 1

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 : $$\begin{align} 3x-y=0 &\iff 3x = y\\ &\iff x = \frac{y}{3} \end{align}$$

Answer 2

Further to the first two comments and Rob's answer:

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 $$\text{“for each real $x,\quad$ <conditions> $\implies$ <implication>”}$$ 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, $2i.$

Answer 3

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: $$\implies\tag{implies}$$

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

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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

Answer 4

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, $$3x-y=0\qquad\to\qquad 3x=y \qquad\to\qquad x = \frac{y}{3}$$

Answer 5

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.)