1. Can someone help me understand contradiction?

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?????

Thank You,
Matt H.

2. Originally Posted by matthayzon89
For example, lets keep it simple, The Difference of any two odd integers is odd.
Do you realize that Difference of any two odd integers is even?

3. Originally Posted by Plato
Do you realize that Difference of any two odd integers is even?

LOL YES! thats my point. The first statement is false. I was wondering if I can use Contradiction to prove it is false by taking the negation of the false statement and proving the negation is true?

4. Originally Posted by matthayzon89
Thats my point. The first statement is false. I was wondering if I can use counter example to prove it is false by taking the negation of the false statement and proving the negation is true?
Well that was not clear.
$7-3=4$ is a counterexample.
It shows that the statement "The difference of two odd integers is odd" is false.

5. Originally Posted by Plato
Well that was not clear.
$7-3=4$ is a counterexample.
It shows that the statement "The difference of two odd integers is odd" is false.

Actually no it does not, I apologize. I confused contradiction with counterexample. I know what counterexample is. Is it possible to prove this problem using contradiction?

I edited all the typo's out so you can re-read it if you would like, it might seem cleaer

Thank you

6. 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.

7. 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?

8. Originally Posted by emakarov
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.

if you consider not p and you find out that not p is true (which is the same as p being false), then you can see that the result of not p proves that p is a cotradiction statement?

9. Originally Posted by Plato
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?

So counterexample is in some sense the negation of a contradiction?

So can you please give me for an simple example of proof using contradiction (without p's and q's lol, i find those confusing)

Thank you, and thank everyone else as well for the replies

10. Originally Posted by matthayzon89
So counterexample is in some sense the negation of a contradiction?
No that is not correct.
You basic problem is your fundamental misunderstanding of terminology.

Originally Posted by matthayzon89
So can you please give me for an simple example of proof using contradiction.
Theorem: The square of an even integer is even.
Suppose that $z$ is an even integer and $z^2$ is odd.
That means that $z=2n$. Which implies $z^2=4n^2=2k+1$.
But that means $4n^2-2k=2(2n^2-k)=1$.
So we have proved the theorem.

11. 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]

Thank you.