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I am looking for a shorthand (or a symbol) for "we have" or its equivalent variants that is not too uncommon. In general I am looking for a notation (shorthand or a symbol) that signals the reader that what follows this sign are quoted facts rather than assumptions or consequents. Just want to increase the efficiency when it comes to hand-written mathematics.

I originally tended to use the shorthand "N.t." for "note that" to fulfill that purpose. However, I learned it is risky in that people not very mathematically minded tend to find it condescending or whatever that compromises the point. So any suggestion that satisfies the desired purpose, even if it is not a shorthand for "we have" (probably a better one), is equally welcomed. Thanks a lot.

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    $\begingroup$ WHTTFCAE (we have that the following conditions are equivalent). No, thanks. ;-) $\endgroup$ – egreg May 16 '18 at 21:00
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    $\begingroup$ @egreg I've seen TFAE frequently for that. $\endgroup$ – Patrick Stevens May 16 '18 at 21:01
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    $\begingroup$ N.B. for Nota Bene? it means observe carefully or take special notice (used in written text to draw attention to what follows). $\endgroup$ – user29418 May 16 '18 at 21:02
  • $\begingroup$ I always write N.t. can you explain a little more about the risk in this phrasing?(maybe I'll also reconsider using it) $\endgroup$ – Holo May 16 '18 at 21:02
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    $\begingroup$ frequently people say "recall" for this. of course that risks condescension too, since it's hard to recall something that you don't know or understand. Similarly the imperative to "note" or "observe" something may irk a reader if that something is new and surprising to them. If you want to be safe, I think using a slightly longer expression such as "we will need the following facts" is a good bet, and all-in-all not so cumbersome. And if you don't want to prove those facts, cite your sources. $\endgroup$ – Badam Baplan May 16 '18 at 21:18
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This is less of an answer and more of a "are you sure your question is the right one?".

You're exactly right about what "note that" means. But why are you looking for a speedy alternative shorthand here?

  • If you're handwriting your own notes, then use whatever shorthand you like.
  • If you're typesetting mathematics, you should always use longhand English.
  • If you're presenting a lecture and therefore writing by hand, I would argue that you should be writing in longhand anyway. Those in the audience who are copying down notes will need more time than you do to absorb what you have just said, and it really helps them if you introduce these little inefficiencies. For example, if they have spent the last thirty seconds trying to understand what you just said, and this has put them behind your board writing, then your "note that" can become "n.t." in their handwritten notes and they have gone some of the way to catching you up.

The major remaining case is that you're writing maths by hand for someone else to read privately; for example, in an answer to a problem sheet question. I'd say that here, too, you should be writing in longhand for your final submission: your job is to argue as persuasively as possible that you have proved something, and your argument is more persuasive the more coherent it is as prose.

Have I missed out your particular use-case here?

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In text you could try so, thus or hence; in a math environment you can use $\therefore$. For equivalence, you can use iff or $\Leftrightarrow$.

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Nota Bene abbreviated "n.b." may be what you are looking for. It was mentioned in the comments. From experience, in only one undergraduate classroom (where rigorous proofs were emphasized), the math professor mentioned the latin phrase and its usage. However, it is not common usage in mathematics, but the relevancy to mathematics exists. For example, see http://platonicrealms.com/encyclopedia/latin-terms-and-phrases-in-math

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