Mastering the Art of Proofs: Tips for Struggling Learners

[462chevelle]

does anyone have any tips on learning how to do proofs. completing a proof isn't something I am good at, at all. I know how to get answers and I can do the math but explaining it to someone step by step, using the right communication. Seems like a foreign language to me. any tips on learning what ways to approach a proof?

 

[lurflurf]

There are several factors. One is considering possibilities. A proof must be general so it must work for all possibilities not just some. You need to know several general approaches like contradiction, contrapositive, induction, direct and so forth. Several things you will want to do is read some common proofs like there are infinite primes and sqrt(2) is irrational and try to understand why the proof works and how someone would come up with it. Practice proving simple things like e^x is not a polynomial and the square of an even number is even.

 

[462chevelle]

the square of an even number is even.
the way I would prove that.
because an even number times an even number equals an even number.
sounds terrible I know..

 

[Mark44]

the square of an even number is even.
the way I would prove that.
because an even number times an even number equals an even number.

Why?

sounds terrible I know..

Any even number n can be written in the form n = 2k, for some integer k. (An odd number can be written in the form 2m + 1.)

What do you get when you square 2k? Can you convince yourself that the result is also even?

There are a couple of books that might be helpful to you: The Nuts and Bolts of Proofs, and How to Read and Write Proofs. You should be able to do a web search on those titles and come up with the authors and other info.

 

[eddybob123]

I read them both

 

[verty]

Eddybob123: Did they work? Are you now comfortable with writing proofs? And can you understand other proofs that you read?

 

[1MileCrash]

the square of an even number is even.
the way I would prove that.
because an even number times an even number equals an even number.
sounds terrible I know..

That's not a proof at all, it's an assertion.

If you set out to prove that the square of an even number is an even number, you start from what an even number is, and what a square is, and show that it is even.

You can't assume that an even number times an even number is an even number, because you are assuming what you are trying to prove in the guise of a slightly more general statement.

 

[462chevelle]

you're right. that is where my language barrier suffers.

 

[HallsofIvy]

One of the things one needs to learn about matheatics is that mathmatical definitions are "working definitions". That is, you can use the precise words of the definitions in problems or proofs.

If I were asked to prove that "the product of two even numbers is an even number", the first thing I would think about is the definition of "even number".

So- do you know what an even number is? What is the definition of an even number?

(Saying that "2, 4, 6 are even numbers" or "even numbers are numbers like 2, 4, 6" are NOT definitions.)

 

[462chevelle]

would you care to define an even number for me?

 

[jbunniii]

would you care to define an even number for me?

It is a number which can be factored as $2n$, where $n$ is an integer.

 

[Edgardo]

Here's a thread with lots of links to material on proofs:
How to write math proofs