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I am leaning this post.

Therefore, I hope you will indulge me and first think of the ‘vectors’ (usually denoted u,v,w,x,y) and their ‘coefficients’ (usually denoted a,b,c) as fundamentally different objects.

in this case, which expression is more appropriate "denoted x,y" "denoted by x,y" "denoted as x,y"?

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Your question will probably be marked duplicate as https://english.stackexchange.com/questions/25179/denoted-by-or-just-denoted, but I think that post didn't give perfect answer.

  • Answers on ELU don't count as duplicates, even if they answer the question. If you want to flesh out this answer, even if you just copy the text from the ELU answer, that would be fine, and I would upvote. However I think the selected ELU answer is too brief (and incorrect), while William Cushing's answer is a bit too comprehensive for this site. Perhaps just paraphrase / combine these two in something an English learner can readily comprehend? – Andrew Jun 27 at 18:31

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