# Displaying a reduction of inequalities

This question is about style and typesetting, but I believe it is more appropriate for this site than a TeX site.

When a bound is being established for some expression, it is not uncommon to see something like this: \begin{align} |\operatorname{Cov}(X,Y)|^2 &= |\operatorname{E}( (X - \mu)(Y - \nu) )|^2 = | \langle X - \mu, Y - \nu \rangle |^2,\\ &\leq \langle X - \mu, X - \mu \rangle \langle Y - \nu, Y - \nu \rangle ,\\ & = \operatorname{E}( (X-\mu)^2 ) \operatorname{E}( (Y-\nu)^2 ) ,\\ & = \operatorname{Var}(X) \operatorname{Var}(Y), \end{align} which I have pulled from the Wikipedia description of the Cauchy-Schwarz inequality.

The issue at stake is whether or not this is correct. Specifically, does a pedantic interpretation of \begin{align} a &= b \\ &\leq c \\ &= d, \end{align} necessarily mean $a = b$; $a \leq c$; $a = d$, or can we argue that the sane interpretation, i.e., $a = b \leq c = d$, is "standard" ... whatever that means.

I think I have let my bias paint this as a silly question, but it comes from a real debate, and there is a reasonable argument for using the interpretation that the aligned relation operators are a shorthand for carrying the left hand side implicitly to each line. If the inequalities are not strict, then you would always be safe using that interpretation, but that is not a good enough reason to adopt that viewpoint. An equality sign gives information about the nature of the simplification that has taken place on that line.

A proposed solution is to fix the alignment: \begin{align} &a = b \\ &\leq c \\ &= d, \end{align} and I guess this might be a good compromise. Certainly it is hard to argue for the commas that appear at the ends of the lines in the Wikipedia example.

A few minutes seaching in the library revealed examples of usage similar to the Wikipedia example in the works of Bourbaki, but even in France, some people don't regard Bourbaki as definitive.

So, I guess that is my real question: is there anything like Strunk and White for mathematics? Is there a definitive answer to the question above? If not, I guess this question should be CW.

Edit

Sometimes people have arguments about what is gramatically correct. This is a moving target, but there are at least certain "authorities" that might be consulted to resolve a dispute. Such disputes cannot usually be resolved by establishing that the sentence in question is unambiguous in its intended meaning, or that such constructions are overwhelmingly common in popular writing. For example, "The Cauchy-Schwarz inequality allows to bound the square of $\operatorname{Cov}(X,Y)$ by the product of $\operatorname{Var}(X)$ and $\operatorname{Var}(Y)$." (I made that up; I did not find it on Wikipedia.) The expression 'allows to' is not (yet) gramatically correct, but it is very common, and its meaning is clear.

The current question is in this vein. I think I can improve the question a bit. I have not looked for a specific example, but I expect I can find an example of a reduction of the form $a = b < c = d$ that is presented as \begin{align} a &= b \\ & < c \\ & < d \end{align}

The intended meaning is again clear, and if that is all that matters, then there is nothing more to say. However, we cannot interpret this in the same way as the previous example. They are not compatible.

I think Qiaochu is correct that there is no authority that can be consulted here, so there is maybe no way for me to salvage the question, but I am hoping to at least show that the intended question is not any more innane than a question about gramatical correctness.

Is my last example actually wrong, or do we just admit that there is no standard interpretation for such a reduction; we are not machines, we are happy to get the intended meaning from the context?

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I don't think there is anything like Strunk and White for mathematics. I think it is clear from context what is meant, and that any other notation is likely to be clunky-looking (but I would love to be wrong about this). –  Qiaochu Yuan Jul 28 '11 at 13:33
You write: "there is a reasonable argument for using the interpretation that the aligned relation operators are a shorthand for carrying the left hand side implicitly to each line". Isn't the question you ask a valid objection to that view? Also, the usual way of using these aligned equations is to take the rightmost side of the preceding line and transform it in some more-or-less obvious way. The ensuing equality $a = d$ (or $a \leq d$) can be quite non-obvious at the end. So I'm curious what that "reasonble argument" would look like. –  t.b. Jul 28 '11 at 13:53
I find it hard to think that anyone seriously could understand such formatting as something other than just line-breaking. (By the way, I've edited the Wikipedia page to remove those ugly commas...) –  Hans Lundmark Jul 28 '11 at 14:41
I don't understand where you get your "pedantic interpretation". I've never seen such interpretation. The fact that the first line "sticks out" to the left is analogous to how in a beginning of a textual paragraph, we indent the first line. The alignment at the equals signs also is a nice stylistic device when the terms on the RHS may run longer than one single line. –  Willie Wong Jul 28 '11 at 16:16
@Theo I see that I did not actually articulate any such argument, and argument is maybe not the right word. The assertion is that the alignment has a semantic meaning. In particular, the claim was that the final equality implies an equality of the lhs and lower rhs. It is clear that this is an unpopular viewpoint. It is also clear that this is not how the author intended the expressions to be interpreted. However, there are at least two people in the universe who believe that this is technically the correct interpretation. One of them is Carl Brannen who answered below. –  yasmar Jul 28 '11 at 17:32

This is a follow up CW "answer" that I can accept in order to put this unpopular question to rest.

The assertion that a sequence of aligned inequalities and equalities, as in the first displayed equation of the question, is incorrect, is completely indefensible. As indicated in the comments, such a position is not popular, but the few comments on this forum are just the tip of the iceberg.

I personally have never seen an issue with this way of displaying a sequence of inequalities, but that mostly means that I had never really thought about it before. However, since I thought the issue was worthy of a question here, and this question was not well received, I feel the need to justify the unpopular viewpoint.

We have these binary relational operators whose standard syntax places them between the first and second arguments. A sequence of manipulations establishes, via transitivity, an inequality between an initial expression and a final expression. This is the qoal: we write $a = b \lt c \leq d = e$ in order to show that $a<e$. The reader receives an explicit sequence of steps for the small book-keeping cost of remembering where our final relation sits in the hierarchy $=,\leq,\lt$. If the steps spanned multiple pages and the hierarchy consisted of dozens of operators, we might think more about how to best display the argument, but that is not the situation.

Now, if our sequence of reductions spans multiple lines, what exactly is the implication of the "stylistic device" that has us horizontally align the binary operators exactly so that the first expression enjoys a privileged position on the left? Does this not emphasise that this is the expression with which we are ultimately making a comparison?

I mentioned in the comments to the question that I expected to be able to find an example where it was evident that the first expression was taken to be the first argument to each of the aligned operators. Well I spent quite a bit of time in the library, and I did not find a single unambiguous example. I looked only in works of authors whose names would be universally recognized, and I found many, many examples by very prominent authors, where the aligned operators must be interpreted as taking their first argument from the end of the preceding line.

I found circumstantial evidence that some do not favour this way of displaying things. There were several works where the need seems to be miraculously avoided, and many where something analogous to a LaTeX "multline" is preferred to "split". However, there are some notable examples where authors explictily avoid the issue being discussed here.

In his "Measure Theory" book, Halmos uses the device of leaving the operator trailing at the end of each line, and then repeating it at the beginning of the next. He does this not only when he aligns the operators (which he does do), but also when he doesn't, and even when only equalities are involved. Thus he would write: \begin{align} |\operatorname{Cov}(X,Y)|^2 &= |\operatorname{E}( (X - \mu)(Y - \nu) )|^2 = | \langle X - \mu, Y - \nu \rangle |^2 \leq\\ &\leq \langle X - \mu, X - \mu \rangle \langle Y - \nu, Y - \nu \rangle =\\ & = \operatorname{E}( (X-\mu)^2 ) \operatorname{E}( (Y-\nu)^2 ) =\\ & = \operatorname{Var}(X) \operatorname{Var}(Y). \end{align} This technique is not unique to Halmos, I also noticed it in the works of Berger and of Banach (and inconsistently with Pólya and Szegö). I think this would probably qualify for what Qiaochu describes as "clunky-looking".

I found another example of a formatting technique that very clearly indicates that the authors see an issue with displaying inequalities this way. In "The General Theory of Dirichlet Series", G. H. Hardy and M. Riesz often display a sequence of equalities with perfect algnment, but when inequalities are in the mix, they increase the indentation so that the subsequent operators are not aligned with the first. Thus they would write something like: \begin{align} |\operatorname{Cov}(X,Y)|^2 &= |\operatorname{E}( (X - \mu)(Y - \nu) )|^2 = | \langle X - \mu, Y - \nu \rangle |^2 \\ & \phantom{(X-\mu)} \leq \langle X - \mu, X - \mu \rangle \langle Y - \nu, Y - \nu \rangle \\ & \phantom{(X-\mu)} = \operatorname{E}( (X-\mu)^2 ) \operatorname{E}( (Y-\nu)^2 ) \\ & \phantom{(X-\mu)} = \operatorname{Var}(X) \operatorname{Var}(Y). \end{align} I think this is quite nice. It is subtle, at least I found it to be so (they maybe used less indentation). If I weren't looking for it, I doubt I would have noticed it.

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I had no problem interpreting the original format correctly and don't feel that there is a need for an alternative.

On the other hand, I agree that it is technically incorrect.

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Thanks for your response Carl. I don't think there is a definitive "technically correct" in this case, so I have posted an alternative "answer", rather than accepting yours. –  yasmar Aug 2 '11 at 9:52