For example, are there set cases when one would write "$x$ is a positive integer" instead of writing "$x \in \mathbb{Z}^{+}$", or vice versa?
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- $\begingroup$ You should do that ... almost all the time .... $\endgroup$Angina Seng– Angina Seng2018-10-04 18:51:21 +00:00Commented Oct 4, 2018 at 18:51
- 2$\begingroup$ It depends on the situation. Some times writing with text is preferred. Other times writing with symbols is preferred. When one should be used over the other is largely subjective, but generally has to do with clarity and aesthetics. $\endgroup$JMoravitz– JMoravitz2018-10-04 18:53:43 +00:00Commented Oct 4, 2018 at 18:53
- 3$\begingroup$ I prefer the $x∈\mathbb Z^+$ notation as I immediately recognize what's happening, whereas the words I have to sit and process them for a little longer to make sure I'm understanding the adjectives correctly. Instead, you'll want to cater to your audience: who are you writing the proof for? Math majors and professionals will all know what you mean, but a liberal arts major may need the words. But more importantly, be smart about readability: $\forall \epsilon>0\exists\delta>0\ni\forall x\in\mathbb D(f):0<|x - a|<\delta\implies |f(x) - L| < \epsilon$ is a garbled mess no one wants to read. $\endgroup$Decaf-Math– Decaf-Math2018-10-04 19:04:09 +00:00Commented Oct 4, 2018 at 19:04
- 1$\begingroup$ Know your audience and no your content and what is clear. I recently answered a post about divisibilty and I said "$x|m$ means$ \frac mx$ is an integer". Now I could have said "$x|m\iff \frac mx \in \mathbb Z$" or I could have said "$x$ divides $m$ means that $\frac mx$ is an integer". I chose to use $x|m$ as we were discussing divisibility and it was already well established that was the topic. It didn't need spelling out. But I chose to say "is an integer" because that was a new relevent purpose of my post. "$x\in \mathbb Z$" would have just be a sideline fact. $\endgroup$fleablood– fleablood2018-10-04 19:49:11 +00:00Commented Oct 4, 2018 at 19:49
- $\begingroup$ One thing is that while $\mathbb Z^+$ is relatively standard, it’s far from universal, and people might wonder if you mean “positive integers” or “non-negative integers,” etc. But writing it out in English doesn’t have that problem. If you’ve already established what $\mathbb Z^+$ means, then using the symbols will make sense for brevity in certain contexts, like if you have to define a bunch of arbitrary positive integers in a proof. $\endgroup$spaceisdarkgreen– spaceisdarkgreen2018-10-04 19:55:05 +00:00Commented Oct 4, 2018 at 19:55
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I will only use notation if it's relevant in some way to the calculation I'm doing or simplifies the presentation of the ideas being discussed. If I ensure that the notation is useful in some meaningful way then I find I can justify using it, otherwise it may simply be clutter.
