/sci/ - Science & Math

>>10220087
Most universities everywhere do this. Thing is it's typically integrated into another course like discrete math or abstract algebra so you don't usually see it called intro to proofs.

>>10220075
There are a lot of discrete math textbooks and a few abstract algebra textbooks that act as intro to proofs books. I'm not aware of any number theory books but that doesn't mean there aren't any.
There are also a lot of intro to proofs books that are just proofs How well they'll work will depend on how much work you put into them.
The intro to proofs course will have a huge effect on your ability to do mathematics going forward (make sure to get a good prof, do not take it with a shitty prof). Our university has an honors version of the course that's taught by a shitty prof and all the students who take it end up struggling a lot more than the ones who take the "easier" course taught by a brilliant prof.
You may also opt to take a formal logic course or to work through a book like The Logic Book by Bergmann (specifically the chapters on Fitch style proof systems).
The idea with proofs is that you have a few basic patterns (see pic related) and that every logical statement, no matter how complicated, can be broken down into a combination of these patterns. So a proofs course will teach you how to deal with each of these patterns. That way, no matter what you're asked to prove you will always be able to break it down and figure out a few strategies to attack it. A number theory book may teach you how to do some proofs but it's really better to actually do an intro to proofs course or text.

That said, Dudley Underwood's 1978 "Elementary Number Theory" second edition book (he has two books with similar names) is a great Number Theory book to work through if you want a straightforward methodical approach.

Wall of text incoming!

>>9409770
It's one of the best intro to proofs books out there. I only have two gripes with it (and 99% of all other books do these too).

1) It never gives a clear and quick run down of all the different proof rules. It's especially annoying because the book almost sets itself up to do so since it contains a (single page) section talking briefly about logical inference (p.61, section 2.11).

So, in place of that please refer to pic related. The rules are laid out in a grid with a row for each 'logical connective' (eg. "[math]\land[/math]") and two columns labeled "introduction" and "elimination". Each rule is then given a short name depending on where it falls in the grid (eg. "conjunction introduction" is "[math]\land i[/math]"), and subscripts are introduced for cells containing more than one rule (eg. "[math]\land e1[/math]" and "[math]\land e_2[/math]").

Each rule contains a space-separated list of assumptions on top, and a conclusion on the bottom (separated by a horizontal bar). It can be thought of as a recipe, eg:
[eqn]
\frac{\text{things I have}}{\text{thing I can make}}
[/eqn]
In short,
>The introduction rules tell you what you require in order to prove a sentence with that connective. Sometimes called "production rules" because they produce something.
and
>The elimination rules tell you what you can produce given a sentence with that connective. Sometimes called "deduction rules" because they deduce something.

(cont.)