I'm just starting out learning how to proof and I am struggling to grasp the basics yet. I'm writing out examples and trying to understand them but I am still not seeing how I should go about doing my own.
This is one example, that I've spent much of today working on:
The sum of an even number and and odd number is odd.
Firstly, I translated this into mathematical language:
$\forall$ x ∈ Z ∃ m ∈ is even ∃ Z ∧ x is odd → m + n = x
Then I've attempted to solve it by using a direct proof, P->Q (I need to prove P, Q is an assumption), however I've got lost here.
I think I need to prove that P (the sum is odd) and manipulate the equation to show that this will always be the case. I'm really lost.
One of the basic tactics for proofs it to write givens at the top of the paper and what you're trying to prove at the bottom. Then work towards each other: sometimes it will be easier to work your way down, and other times it will be easier to work backwards and work your way up. Once you've connected the top and bottom, you're done.
Givens include the conditions (that is, the "if" part of an "if then" statement), axioms, definitions, and applicable theorems. Givens can also take other forms, for instance if you're asked to prove "For all X, Y is true", it's given that you're dealing with X.
So here, your first givens are:
m is even
n is odd
The next step is to go through all the key words and find their definitions. The two key words are "even" and "odd". So write down what the definitions of "even" and "odd" are. Usual definitions are:
there is i such that 2i = m
there is j such that 2j+1 = n
So now you can take the expression m+n and substitute in: m+n = (2i)+(2j+1)
If you've written what you're supposed to prove at the bottom, you should have "(m+n) is odd". You can use the definitions above to say that this is equivalent to "there is k such that 2k+1 = m+n". You can then write that right above your bottom statement.
So now you just have to connect m+n = (2i)+(2j+1) to m+n = 2k+1.
So:
Write down what you're told at the top (conditions, etc.)
Write down what you're supposed to prove at the bottom
If you can think of any theorems that apply, write those down at the top, just below what you're told
Search for key words in what you've written down so far
Find the definitions for key words, and write them at the top, just below any theorems
Work down from the top and up from the bottom until they meet