I'm a mathematics major, but I'm not native in English and I have some questions about the "terms" used in mathematical proofs.
What's the difference between "It is suffice to show that" and "It is enough to show that"? is it exactly the same thing? "Enough To Show", "Want To Show", "Need To Show" seems all the same to me..
Does "Suppose that", "Assume that", "Let" and "If" have all different nuances?
Let $A \subset \mathbb{N}$. Then $A$ is countable.
Suppose that $A \subset \mathbb{N}$. Then $A$ is countable.
If $A \subset \mathbb{N}$, then $A$ is countable.
Assume that $A \subset \mathbb{N}$. Then $A$ is countable.
Doesn't these sentence all carry the exact same nuances?
I know that these questions are really meaningless, but I would love to know if there are differences or not!
This is an English question but it is English specifically in the context of mathematics I guess.
Suffice and Enough are the same thing. But "Want to show" is different. It will frequently be used for the same meaning as those two, but it explicitly declares something as your objective, while "it suffices" and "It's enough" only weakly imply that showing it is your objective.
Suppose and Assume are pretty much the same although a supposition is less likely to carry on throughout the entire piece of work - it may be relaxed later. An assumption can later be relaxed too, but there's a suggestion more permanence to an assumption - suggesting it may be assumed through to the end.
Again, "Let" will almost always be a permanent assumption - carrying through to the end of the paper. If X on the other hand, should really have its end demarcated by an if not, or otherwise, or a irrespective of X. An open-ended if statement is ideally to be avoided, but in reality the end of the if may well be clear in its context.