I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and logic (or lack thereof), I noticed an issue that I wasn't as aware of before this experience. A lot of students used English words incorrectly (from a mathematical point of view) when writing sentences within their proof.
Some of the words/phrases I am referring to are:
- Such that
- There exists
Is anyone aware of a reference which explains how to use these words and phrases (and others that frequently occur) in a mathematical context?