Is it acceptable style to mix equalities and inequalities in one line

Question

Is this considered bad style

$$2 = \sqrt{4} < \sqrt{16} = 4?$$

It seems as though this is not strictly correct, since $2 = \sqrt{4}$ is a logical proposition which represents boolean value (true or false). A boolean value cannot be less than $\sqrt{16}$.

On the other hand, I am sure that most people will correctly interpret this as shorthand for $2 = \sqrt{4},$ $\sqrt{4} < \sqrt{16},$ and $\sqrt{16} = 4$

Answer 1

If $R_1, R_2, \ldots$ are binary relations, it is standard practice in mathematics to write:

$$a_1 \mathrel{R_1} a_2 \mathrel{R_2} a_3 \ldots a_{k} \mathrel{R_{k}} a_{k+1}$$

as a short hand for:

$$a_1 \mathrel{R_1} a_2 \mbox{ and } a_2 \mathrel{R_2} a_3 \ldots \mbox { and } a_{k} \mathrel{R_{k}} a_{k+1}$$

This convenient syntactic convention works because, in most mathematics, we usually write as if we are working in first-order logic, where boolean values aren't allowed as operands of relation symbols: $(1 < 2) = (3 < 4)$ isn't allowed. When you work in higher-order logic in mathematics, and in most programming languages, formulas like $(1 < 2) = (3 < 4)$ are allowed and so this convention doesn't work so smoothly.

Answer 2

You will find such things often on this forum, and the meaning is usually clear. When I was an undergraduate we used to string $\implies$ between statements we had shown were dependent on previous statements - it was convenient shorthand rather than boolean logic.

The purpose of mathematical writing is to communicate clearly and accurately what you mean - that can depend on context and audience. If shorthand brings in an ambiguity, then write it a different way.

Answer 3

It's fine - people write that way all the time. But don't ever do this: $$1\le b=c>d.$$

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Edit: Various people have commented, saying that there's nothing wrong with the above. Perhaps not; it bothers me, but I'm not going to insist that it's wrong. If I claimed I didn't actually say it was wrong people would say I was being pedantic.

One person points out that if you write the above it certainly is wrong to deduce a relationship between $1$ and $d$. And that's the problem - in my experience in "beginning analysis" classes students who write things like what's above do tend to draw incorrect conclusions. So I'm going to just rephrase what I said: "Wrong or not, don't do that. It's a bad idea."