For example, lets keep it simple, The Difference of any two odd integers is odd.
So would contradiction proof example be something like (the negation of the statement above)...... The difference of any two odd integers is even and then prove that the negation is true, meaning that the original statement is false?????
Proof by contradiction works like this. You need to prove P. Instead, you consider (not P). Then you prove that this (not P) implies a contradiction. Therefore, the original P is proved to be true.
If you need to show that the difference of two odds is even, you fix two arbitrary odd numbers x and y (so far the proof is like for every method) and assume that x - y is odd. From this assumption you derive contradiction, such as 0 = 1. Therefore, x - y being odd is false, i.e., x - y is even.
Well now I understand your confusion.
Have you ever heard the expression, You cannot prove a negative?
Well it is true. We do not prove a statement is false.
That is the purpose of counterexamples.
We use a counterexample to show that a statement is false.
On the other hand, proof by contradiction is commonly used to show that a statement is true.
Does that distinction make sense?
You basic problem is your fundamental misunderstanding of terminology.
Proof by contradiction.
Suppose that is an even integer and is odd.
That means that . Which implies .
But that means .
That is a contradiction.
So we have proved the theorem.
Okay, I get it now
So basically if you are trying to prove:
a,b, and c are any integers. if a|b and b|c then a|c by using contradiction you would say something like:
if a|b and b|c then 'a' DOES NOT divide 'C', and then you would go ahead and try to prove this negation 'as if' it was true until your reach a point of contradiction which proves the original statement that says a DOES divide c.
[in your head or on paper you come to a realization that this statement true before you start working on proving it and then you try to solve it as if it was false to prove the 'non-believers' that their way of thinking is illogical and they are wrong]