TITLE: Use of $\implies$ and $=$ when stricter conditions are true
QUESTION [7 upvotes]: I am about to begin studying as an undergraduate. Previously, the mathematical exams I have studied have been incredibly lax in terms of notation or neatness of proof - it would not be penalised, for instance, to write $f'(x)$ instead of $\frac{dy}{dx}$ when the function given was in terms of $y$. It was also considered equally valid when given an identity to prove, to work from the LHS and then from the RHS and join the two together in the middle (rather than a coherent working from one side to the other).
Anyway, as it is not something featured in my previous syllabus, I am looking to improve my understanding of notation to use when proving a statement. I have come across two inconsistencies in teaching materials so far:
1. Use of $=$ when $\equiv$ is valid.
Now, I am comfortable with the meaning of these symbols. But my question is when the use of $\equiv$ or $=$ is mandatory, and when it is permissible. An obvious case to me would be something like: $$\frac{3x^2+34x+6}{x^2+2x-24} \equiv \frac{3x+7}{x-4} + \frac{9}{x+6}$$ Here, use of $=$ would imply it is an equation to solve, rather than something true $\forall x \in \mathbb{R}$ ($x \ne -6, 4$). All materials I have come across would use $\equiv$ here.
But why is it common to write $3+2=5$, when clearly this is not an equation to solve, but an equivalency? Would $3+2 \equiv 5$ not be more accurate? In a more borderline case, if part of the answer to a question includes the simplification of $3(x-4)+2$ to $3x-10$, which symbol is most appropriate? I have seen both $ 3(x-4)+2 = 3x-10$ and $3(x-4)+2 \equiv 3x-10$.
The definition of the $\equiv$ symbol (two things are equal for all values of any variables used) seems to contradict the notation I have seen in common use, based on its lack of use outside of proving identities.
2. Use of $\implies$ or $\impliedby$ when $\iff$ is possible.
If I am writing a proof and we progress from one stage to the next, it seems that $\implies$ would demonstrate the flow of logic more accurately, even if $\impliedby$ is also true and thus $\iff$ could be used. For example, if a question asked us to find all possible values of $x$ in some trigonometric equation, and we had simplified down to:
$$\sin(x) = 0 \implies x = n\pi\; (n \in \mathbb{Z})$$
Here, I am conflicted, because $\implies$ seems to be more coherent, as we are interested in progressing from the left to the right but have no use for the information that progressing from the right to the left is valid. However, if $\implies$ contains within it the implicit notion that $\require{cancel} \cancel{\impliedby}$, then the use is incorrect and should be replaced by $\iff$. Which symbol should be used, or are both permissible (and the question down to personal preference)?

REPLY [8 votes]: It is absolutely fantastic that you are thinking about these issues as you start your undergraduate career. If only more students would appreciate these subtleties in mathematics. One nice thing in mathematics is that everything can be made precise.
A couple of remarks though:

Often we will define precise use of notation and then immediately violate the convention. Insisting on precise notation everywhere is not helpful because everything becomes too cumbersome. It can become easy to bury what you are trying to communicate in notation. Abuse of notation is very common and accepted. Also, mathematics is about more than a game of notation. While it strictly speaking might be true at the root, mathematics is also about ideas.
Remember the context. Saying that $x^2 + x  + 1=0$ might mean an invitation to solve the equation. It might mean that $x$ is an elsewhere defined number and that the number $x^2 + x + 1$ is equal to $0$.
Ask your teacher. The teacher will probably have certain preferences when it comes to notation. Don't then get mad at your teacher because he/she doesn't follow notation that you have used before. Instead, get used to the change of preferences.

Now, typically $=$ is used to say that two elements in a set are the same element. Saying "solve $x^2 = 2$" can then mean find all elements $x$ in the set $\mathbb{R}$ whose square is the same as the element $2$ in the set of real numbers. Saying that $2(x-3) = 2x -6$ might say that the polynomial $2(x-3)$ is the same as the polynomial $2x - 6$. It might mean that the function $f:\mathbb{R} \to \mathbb{R}$ given by $f(x) = 2(x-3)$ is the same (as element in a set of functions) as the function $g:\mathbb{R} \to \mathbb{R}$ given by $g(x) = 2x - 6$.
Saying that $\frac{x^2}{x} = x$ might again be about an equality of functions. But what it the domain of these functions? both functions would have the same domain, namely $\mathbb{R}\setminus\{0\}$.
You say that "if $\implies$ contains within it the implicit notion that $\require{cancel} \cancel{\impliedby}$ ..." This is simply not true. You can, in fact take the definition of $A\iff B$ as ($A\implies B$ and $B\implies A$). So indeed both of the following is correct
$$
x^2 = 1 \implies x\in \{\pm1\} \\
x^2 = 1 \iff x\in \{\pm1\} 
$$
Both are therefore permissible.
Here is the thing. When you are writing a proof you want to be careful and precise. Getting into the habit of writing $\iff$ everywhere you can will likely lead you to use it wrongly at some point. Being careful is to prove that $A\iff B$ by first showing $A\implies B$ and then $B\implies A$ even if you could do both at the same time.
The symbol $\equiv$ is often used in different ways. It will often depend on the definition. I think that most sources would not use $\equiv$ in the place of $=$.