Use of $\implies$ and $=$ when stricter conditions are true

Question

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)?

Answer 1

Ad 1. You write

But my question is when the use of ≡ or = is mandatory, and when it is permissible.

There is no absolute answer to this, yet the use of '=' between terms like the one you give, which can be seen as a elements of the 'rational function field' $\mathbb{Q}(x) = \mathrm{Frac}(\mathbb{Z}[x])$ is much more usual in modern mathematics. The use of $\equiv$ you have encountered seems to be some sort of 'subcultural' usage confined to some schoolbooks (I guess).

To explicitly address the example you give: I can assure you that most mathematicians would consider

$$\frac{3x^2+34x+6}{x^2+2x-24} \equiv \frac{3x+7}{x-4} + \frac{9}{x+6}$$

an unusual notation. The most usual notation is to write

$$\frac{3x^2+34x+6}{x^2+2x-24} = \frac{3x+7}{x-4} + \frac{9}{x+6}$$

and interpret this as an equality in the field $\mathbb{Q}(x)$ of rational functions. Yes, in the background, depending on your choice of set-theoretic foundations, there may be equivalence classes somewhere, yet this is irrelevant, and this is a genuine equality.

I would appreciate if some other people would second this (uncontroversial) opinion and talk the OP out of their fear of using = here. There might be good reasons to use other symbols for this sometimes, but recommending the OP to use whatever they see fit is definitely misleading.

In particular, your statement

All materials I have come across would use ≡ here.

if true, implies that you have been brought up on a very limited set of 'materials'.

And in

But why is it common to write 3+2=5, when clearly this is not an equation to solve, but an equivalency?

the statement about the "equivalency" is wrong, at least relative to usual contemporary mathematics. This is an equation between natural numbers.

As a general rule, remember that

• when you are working with the elements of an algebraic structure (group, ring, field, ...), then the usual 'relation symbol' is '=' and not $\equiv$.

The symbol $\equiv$ has various meanings in various contexts.

Ad 2.

Re

However, if ⟹ contains within it the implicit notion that

Please note that the usual convention is that $\Rightarrow$ never means $\require{cancel} \cancel{\impliedby}$. Never. If it would, it would be impossible to express $\Leftrightarrow$ in terms of $\Rightarrow$ and $\Leftarrow$.

Re

Which symbol should be used, or are both permissible (and the question down to personal preference)?

The latter. Both are permissible. The choice is a choice of emphasis. And of course they do not mean the same, as you know.

Does that help?

Answer 2

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:

1. 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.

2. 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 $0$.

3. 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 $=$.

Answer 3

It's nice that you have this disquietude at this point.

Mathematical precision and clarity depends on the context, the writer, the intended reader and their common backgrounds.

Absolute precision using just symbols is possible in theory but in practice you will be much better using words instead.

Hence, whether the symbol "=" means an equation (so you are talking about whether there is some $x$ for which equality holds), a statement (so $x$ has been fixed before and you are asserting that both sides of equality are the same number), an identity (so for all $x$ in a certain set the resulting numbers on both sides of the equality will be the same number), or a functional equality (so the functions implicitly defined by both expressions -which is often an imprecise way to define a function- are the same function), etc, should either be clear from the context or it should be made clear on the fly.

When solving an equation such as $\sin(x)=0$, you should use <--> because you want to describe the entire set of solutions, not more, not less.

Now if you are proving an implication with a chain of implications, you'd normally use words instead that of symbols. In any case, you will assert that two things are equivalent instead of just saying that one implies the other if this hint will produce a better impact on the reader. For the logical validity of a proof, if you show that a-->b-->c-->d then you have shown that a-->d, and that's all you need to argue. But telling the reader that actually b<-->c can make their life easier or harder, depending on the context.

In summary, good math writing is not about utmost symbolic precision, it's about understanding who you're writing to.