How to do math proofs?
Im taking Abstract Algebra next semester and I was wondering if you guys can explain the concept of how to prove things. My problem is when you're given a statement, how do you know where to begin and what method to use. I know from a class that you took you can use the backward method working from the conclusion and trying to connect your hypotheses and such. If you guys can provide some examples so I can get a good visualization on how this works I would greatly appreciate it.
Also did you prefer Algebra or Analysis?
Comments
It's good to see you asking a lot of questions about these transition courses. Motivation is the largest player on the road to successfully learning proofs and appreciating higher level mathematics.
I mentioned in an earlier response that grasping definition and applying them correctly is the key to success in introductory level classes. Here is a primitive example.
Definition: A "natural number" is a non-negative whole number. For example, the first few natural numbers are: 0,1,2,3,4,5...
Definition: A natural number n is said to be "even" if there exists a natural number k such that n = 2k
This is a familiar concept from grade school, but it is formally defined here. It says that a natural number is even if and only if it can be written as 2 times another natural number. Any time we want to show that a natural number is even, we must show that our number is 2 times a natural number. For example we can show that 18 is even by noting that 18 = 2*9, 168 is even because 168 = 2*84, etc.
Below is a simple proof that uses this definition. I will write out the proof first and then walk through it after.
Theorem: If n is even then n^2 is even.
Proof: Suppose n is even. Let k be a natural number such that n = 2k. Then n^2 = (2k)^2 = 4k^2. Consequently n^2 = 2(2k^2), and therefore n^2 is even.
Walkthrough: We wanted to show that the square of an even number is even. To show this we chose an arbitrary even number n. Our entire goal now is to show that n^2 is even. We applied the definition of what it meant for n to be even; this allowed us to find a natural number k such that n = 2k. We looked at n^2 and simplified it to n^2 = 4k^2. We have hopes of writing n^2 as 2 times a natural number since this will prove that n^2 is even. We did so by factoring out a 2 and writing n^2 = 2(2k^2). Since 2k^2 is a natural number and n^2 is 2 times 2k^2, it follows by definition that n^2 is even. We have then shown that if n is even that n^2 is even.
The basic format for this theorem is an "implication" it is of the form: If P then Q. Where P is the statement "n is an even number" and Q is the statement "n^2 is an even number". One way to go about proving a theorem in the form of an implication is to assume P is true. The goal is to step through a series of sentences and conclude at the end that Q is true. This will show that if P is true then Q is true. This method is referred to as proving "directly". There are other methods to prove implications. Another one of note is called "proof by contrapositive".
This is just one example. It is one of the simplest types of proofs. I can't fit much in due to the limit on post lengths. I hope this example helps. Feel free to contact my e-mail through Yahoo Answers if you have any specific questions.
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