It is absolutely fantastic that you are thinking about these issues as you start your undergraduate career. If only more students would appreciate these subtleties in mathematics. One nice thing in mathematics is that everything can be made precise.
A couple of remarks though:
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Often we will define precise use of notation and then immediately violate the convention. Insisting on precise notation everywhere is not helpful because everything becomes too cumbersome. It can become easy to bury what you are trying to communicate in notation. Abuse of notation is very common and accepted. Also, mathematics is about more than a game of notation. While it strictly speaking might be true at the root, mathematics is also about ideas.
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Remember the context. Saying that
manpreet![Tuteehub forum best answer]()
Best Answer
2 years ago
I am about to begin studying as an undergraduate. Previously, the mathematical exams I have studied have been incredibly lax in terms of notation or neatness of proof - it would not be penalised, for instance, to write