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Here is the same proof formatted two different ways. My questions is what is the standard for a writeup like a homework assignment? I think the first one looks better but I don't think that is the way that I normally see it. If someone could just say which one is better I would really appreciate it.

Line by line not line by line

Jac Frall
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    I’ve seen the second more in textbooks, probably because they’re trying to save space. I’ve always written my homework more like the first. – Joe Sep 26 '19 at 00:04
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    When writing in the second style, my advice is to try to vary the phrases to avoid repetition and also avoid too many hidden “then’s” (for example “since $x\in A_1$ then $x\in A_1 \cup A_2$” is clearer than “since $x\in A_1, x\in A_1 \cup A_2$” although the latter is common since it is shorter, but is potentially confused for a list of hypotheses rather than an “if-then”) – Nap D. Lover Sep 26 '19 at 00:46
  • @NapD.Lover that is good advice – Jac Frall Sep 26 '19 at 01:07
  • The second one screams "Wall of text" to me. – Giacomo Alzetta Sep 26 '19 at 15:19
  • I believe the first is the most common used in proofs in Physics text books. I've seen the second used more frequently in math, perhaps because there are often more steps. – TurlocTheRed Sep 26 '19 at 15:21

3 Answers3

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How a proof is written up depends on the purpose and the audience.

Your example looks like an exercise in a first course in discrete mathematics asking you to demonstrate some elementary formal reasoning about sets and functions. The instructor (or grader) should be able to check your argument without struggling with formatting. So the first version is clearly better than the second.

The first one would be even better with some blank lines (paragraph breaks) between sections of the argument, and a few more words telling the reader what part of the argument is next.

In textbooks and in research articles much more of the argument in a proof will be in words. It's the ideas that are important., The author will assume that the reader can produce a more formal proof.

Ethan Bolker
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The first one is way easier to read. I'm always telling off students who squash all their work up into a tiny space. Apart from reading it, it's also much easier to write comments on specific bits when it's spaced out. Paper is not expensive any more, so there's no reason to cram everything in.

If you must do the second, at least double-space the lines and put space between paragraphs.

People often say that papers are written like the latter. This is true, but people who are writing papers hopefully know what they're doing, and have been in the business a lot longer than people writing homework. Whereas on homework, you are expected to get things wrong, or inaccurate, or miss things out: the whole point of having someone look at it is to correct these things. (Even PhD theses are usually set double-spaced.)

Chappers
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    I don't think Ph.D. theses are set double spaced so that people can make corrections. I think it comes from rules written down when theses were prepared on manual typewriters. – Matthew Leingang Sep 26 '19 at 09:57
  • “Paper is not expensive any more” – more importantly, nobody should ever print proofs on paper. (I know, many mathematicians still disagree.) – leftaroundabout Sep 26 '19 at 15:29
  • “nobody should ever print proofs on paper” Have you ever read a paper? As in “reading a paper”, not as in “reading” a paper. – Lukas Juhrich Oct 01 '19 at 18:34
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I would opt for the second style, with slight variations. I would have formulated the two statements that need to be proved such that one uses $\subset$ and one uses $\supset$. Then I would have made each of the proofs of these two statements a paragraph and added headings "$\subset$" and "$\supset$", including the quotation marks.

Carsten S
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