The wording of proofs

Dear Sir,

This may seem like a stupid question, but it has been bugging me for ages..

Is there a difference between me saying there are or there exist?

Similarly, is suppose equivalent to let and if? Also, is hence equivalent to therefore, so, i.e. ?

What I mean by “is there a difference” I mean could I lose marks in an exam if I write something equivalent?

Reply:

The short answer is this: If you write a clear and logical proof, you will never have marks taken off for using one phraseology or another. The point is that when you read proofs in a book or lecture notes, you should understand precisely what’s meant by the sentences used, rather than trying to memorize the precise adverbs, prepositions, etc. Now of course, since mathematical language depends on conventions like any other language, you should eventually learn them naturally . But it’s safe to say that in a proof, the primary focus should be on correctness and clarity . To achieve this, the most important point is obviously understanding .

To answer your specific questions:

-I can’t think of any particular situation where there would be a difference between `there are’ and `there exist’.

-`Suppose,’ ‘let,’ and, ‘if’ are used in different contexts so I can’t answer this question without some sample sentences you have in mind. Perhaps you are thinking of sentences that begin

`Let X be…’
`If X is…’
`Suppose X is…’

in which case they seem to be mostly equivalent.

-‘Therefore,’ ‘hence,’ and ‘so,’ are interchangeable in any mathematical sentence I can think of where they are used as  connectives leading to a conclusion. ‘Hence’ and ‘so’ are a bit more informal, while ‘therefore’ might be more likely to occur in final conclusions. ‘i.e.’ is usually different. As in common parlance, it means ‘that is,’ and is used mostly to just rephrase something.

If your sensitivity to differences in the turns of phrases arises from a lack of confidence in following mathematical arguments, you should definitely take it seriously. You need to make your understanding immune to insubstantial differences. (In particular, you should have a good sense of which differences are ‘insubstantial.’)

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