# Proof Practice

1. Feb 6, 2016

### UncertaintyAjay

Could anyone recommend some books or exercises to practice proofs? Or even post some theorems to prove?
Ajay

2. Feb 6, 2016

### WWGD

I am not sure of what you are looking for, but most elementary books contain proofs, and contain theorems. Look for some theorem that interests you, try to prove it and if you're stuck, just post it here and we'll gladly help.

3. Feb 7, 2016

### MostlyHarmless

Also, if you scroll through the Calculus and beyond section you might find some exercises to try to prove.

Or.. you might prove that there are no positive integers $a,b,$ and $c$ such that for $n \geq 3$

$$a^n + b^n = c^n$$.

4. Feb 8, 2016

Haha.

5. Feb 8, 2016

### Krylov

What field of mathematics and what level are you interested in?

6. Feb 8, 2016

### UncertaintyAjay

I'll be headed to university this fall. I am familiar with calculus, really basic number theory, geometry, algebra etc.

7. Feb 8, 2016

### MostlyHarmless

You might start by trying to prove some statements about parity. Like an odd number times an odd number is always odd. Even number times anything is always even. Those are pretty straight forward. If you feel you already feel comfortable with those. Then really you should just find a good introductory proofs book, and begin familiarizing yourself with basic proof strategies.

8. Feb 8, 2016

### UncertaintyAjay

Yeah, I've done those. I am also familiar with a few proof strategies like induction and proof by contradiction but concepts are a bit hazy. Any recommendations for books?

9. Feb 8, 2016

### Krylov

Ok. The following is perhaps more about notation and careful reading than about mathematics proper, but here it is:

Prove or disprove: $\{x > 0 \,:\, x < r\,\forall\,r > 0\} \neq \emptyset$.

Does that mean that you know what a countable set is?

10. Feb 8, 2016

### UncertaintyAjay

I assume this is for x and r both belonging to real numbers?

Assume the above statement to be true. Then for any element of the set S {r:r>0} there must be another element smaller than it. I.e there is no smallest element of S.( Now i am a bit stuck, I know that there is no smallest element in the set of real numbers, so the statement is true,but I haven't the foggiest idea how to prove it. I'll think over and see if there is another approach to the proof as well)

11. Feb 8, 2016

### UncertaintyAjay

I do. The set of natural numbers,prime numbers, odd numbers, even numbers etc. are countably infinite. Real numbers, irrational numbers etc. are not.

Edit: Sets like "Natural numbers less than 10" are countable. Just not countably infinite. Uncountable sets are ones where you cannot create a one to one function with the set of natural numbers.

12. Feb 8, 2016

### micromass

Staff Emeritus

13. Feb 8, 2016

### UncertaintyAjay

Countable. Right?

14. Feb 8, 2016

### Krylov

Yes, that was implicit in the $>$ sign, but you are very correct.
Ok
Beautiful, then I already have some other calculus-flavored exercises in mind.

15. Feb 8, 2016

### UncertaintyAjay

Bring it on. I haven't been this excited in ages.

16. Feb 8, 2016

### Krylov

I was thinking about this one, but it may be a bit too difficult, since a number of steps are required. However, in principle it should be doable.
1. Let $f : [0,1] \to \mathbb{R}$ be nondecreasing. Prove that $f$ has a countable number of discontinuities.
2. Let $g: \mathbb{R} \to \mathbb{R}$ be nondecreasing. Prove or disprove: $g$ has a countable number of discontinuities.
Take your time for it. (Don't feel obliged either. If others have exercises that appeal more to you, go ahead and do those.)

17. Feb 8, 2016

### axmls

I highly recommend Spivak's Calculus. I found it to be extremely well-written, and fostered a love of pure math in me (an engineering student).

18. Feb 15, 2016

### 3301

Maybe this can help with proofs: fb/brilliant
Here you can do this problem:

Find the prime number which is one less than a perfect square number?