Proof Practice Ideas: Books, Exercises, & Theorems

[UncertaintyAjay]

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

 

[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.

 

[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 $n \geq 3$

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

 

[UncertaintyAjay]

Haha.

 

[S.G. Janssens]

Or even post some theorems to prove?

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

 

[UncertaintyAjay]

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

 

[MostlyHarmless]

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

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.

 

[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?

 

[S.G. Janssens]

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

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

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

I am also familiar with a few proof strategies like induction

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

 

[UncertaintyAjay]

{x>0:x<r∀r>0}≠∅{x>0:x<r∀r>0}≠∅\{x > 0 \,:\, x < r\,\forall\,r > 0\} \neq \emptyset.

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)

 

[UncertaintyAjay]

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

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.

 

[micromass]

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

What about rationals?

 

[UncertaintyAjay]

Countable. Right?

 

[S.G. Janssens]

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

Yes, that was implicit in the $>$ sign, but you are very correct.

I'll think over

Ok

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

Beautiful, then I already have some other calculus-flavored exercises in mind.

 

[UncertaintyAjay]

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

 

[S.G. Janssens]

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

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.)

 

[axmls]

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

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).

 

[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?

For more view this link