Repeat $\le,\subset$ when starting a new line?

Question

My question is very simple. When one sequence of inequalities or inclusions is broken into two or more lines, one should repeat the symbol $\le $ or $\subset$ when starting a new line?

In particular, suppose we want to write $A_1\subset A_2\subset A_3\subset A_4$ in two lines, then the correct notation is

$$A_1\subset A_2\subset $$ $$A_3\subset A_4$$

or the following

$$A_1\subset A_2\subset $$ $$\subset A_3\subset A_4$$

Also, one should write

$$1\le 2\le $$ $$3\le 4$$

or

$$1\le 2\le $$ $$\le 3\le 4$$

Thank you!

Answer 1

There is nothing like IUPAC for maths. Read recent math books to learn how to write maths now.

My suggestion: the duplication of signs like $\subset$, $\le$, $=$, and so on is useful to make clear that the former line is not an end and the latter is not a beginning, specially when you write maths by hand. Indentations are widely used today, and ending signs are omitted.

Answer 2

If you don't have the $\le$ or $\subset$ at the start of the second line, it can be quite confusing to the reader as it isn't immediately clear how this line is related to the previous one. I would tend to use $$ \eqalign{ A_1 &\le A_2 \cr & \le A_3 \cr & \le A_4} $$

Answer 3

Note you can hit "edit" to see how things are done in the answers.

To add a bit to Robert's (correct!) answer and Michael's (correct!) comment:

First, don't ever do it the way you did in your question, with two separate displays! You get unpredictable results.

Of course if it's actually $A_1\subset A_2\subset A_3\subset A_4$ that should be just one line. When you do have a chain of inclusions or inequalities or whatever that needs to be broken into several lines you want to use align. And you want to use the & to align the relation symbols. (As in Michael's comment: the two $\le$ are both directly beneath the $=$.)

I think "everyone" agrees so far. Here's my opinion about an issue that comes up. I used to write like this: $$\begin{align} \left|\int_0^1\sin(t+t^2+t^3+t^4)\,dt\right|&\le\int_0^1|\sin(t+t^2+t^3+t^4)|\,dt \\&\le\int_0^1|\sin(t+t^2+t^3+t^4)|\,dt \\&\le\int_0^1|\sin(t+t^2+t^3+t^4)|\,dt \\&=\int_0^1\,dt=1,\end{align}$$ feeling there was no reason to add another break. These days I tend to feel that if every expression has its own line then every expression has its own line - I write $$\begin{align} \left|\int_0^1\sin(t+t^2+t^3+t^4)\,dt\right|&\le\int_0^1|\sin(t+t^2+t^3+t^4)|\,dt \\&\le\int_0^1|\sin(t+t^2+t^3+t^4)|\,dt \\&\le\int_0^1|\sin(t+t^2+t^3+t^4)|\,dt \\&=\int_0^1\,dt \\&=1\end{align}$$ instead, to be consistent. Easier to read(?), because you know that the next thing is below, not maybe below and maybe to the right, depending.