In proofs, is 'write' equivalent to 'let' or 'suppose'?

Question

In my textbook, the proof of the statement "If A = [a b c d], then..." starts with 'Write A = [a b c d]...'.

Is this similar to 'Let A = [a b c d]...' or 'Suppose A = [a b c d]...'?

Answer 1

Yes, "write" is about the same as "let" or "suppose". Using "write" instead of just plain "let" sometimes occurs when the author is establishing notation as well as naming something to be used in the text that follows.

Answer 2

IMHO the most important thing to understand is that, essentially, there are no rules for writing proofs. (Note: I am specifically referring to the writing, not the content.) As long as the proof makes your arguments clear to the readers for whom it is intended, nothing else matters very much. There is absolutely nothing wrong with having a personal style in mathematics, as in any other kind of writing.

Having said that, let me say briefly how I myself would use these words - with no expectation that you must use them in the same way. I would normally tend to say "write" in order to specify notation, or something else which could easily be altered with no essential difference to the proof. I would say "let" or "suppose" when I am actually making a substantive assertion.

Let $n$ be even. Then we can write $n=2k$, where $k$ is an integer.

The "Let" at the beginning is not really about "$n$", it is about "even". If $n$ were odd instead of even, that would be an important difference. So I would not use "write" here. On the other hand, the "write" is introducing the variable $k$. If I changed $k$ to some other variable, say $m$, it would make no serious difference. But the terms are to some extent interchangeable: I would see nothing wrong with the following.

Suppose that $n$ is even, and let $n=2k$, where $k$ is an integer.

The one place where I would have a definite preference for "suppose" is at the start of a proof by contradiction, where we are assuming something which eventually we will prove to be false.

Suppose that $\sqrt2$ is rational.

And to finish, let's continue the previous example in three ways which IMO are equally good. I'll use "write", or "let", or nothing at all.

Suppose that $\sqrt2$ is rational. Then we can write $\sqrt2=p/q$, where...

Suppose that $\sqrt2$ is rational, and let $\sqrt2=p/q$, where...

Suppose that $\sqrt2$ is rational. Then $\sqrt2=p/q$, where...

Hope this helps!