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## #1 2013-10-13 10:21:46

Al-Allo
Member
Registered: 2012-08-23
Posts: 324

Hi, I have a little question concerning contradictions :

If I have a statement "A" that I want to prove, and only have the possibility for it to be True or False.

After some manipulations, I arrive at some contradiction. (Here's where my question begins.)

How can we know that a contradiction is enough to be sure at 100 % that a statement is not correct?

Is it because in Mathematics, for a thing to be True or False, it must always be ALWAYS "working" without arriving at some contradiction ? (Mathematical ideas must always work, and not sometimes yes, sometimes no.)

I just want to be sure of thinking of it in the right way, corrections would be greatly appreciated ! Thank you !

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## #2 2013-10-13 10:28:03

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

A contradiction is by definition a statement that is always false. If by manipulating you find that a statement is equivalent to some contradiction, it must be equivalent to false and thus itself is false.

Here lies the reader who will never open this book. He is forever dead.
Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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## #3 2013-10-13 19:27:32

bob bundy
Registered: 2010-06-20
Posts: 8,340

hi Al-Allo

In mathematics, if you start with a TRUE statement and use correct working to obtain another statement, then that statement must also be TRUE.

So, let's assume A is FALSE.  If we can do correct working that leads to a statement that is FALSE, then the assumption must have been incorrect.  That's how we know A is TRUE.

Simple example.  Let's use reductio ad absurdum  to prove that for any integer, n, 2n is EVEN.

A = "2n is EVEN, where n is an integer"

Assume this is FALSE.  ie. 2n is not EVEN.

This means that 2n is ODD.

This means that 2n + 1 is EVEN, so it can be divided by 2, giving an integer result.

(2n + 1)/2 = n + 1/2. This is an integer.

Subtract n (an integer) and the result is also an integer

This means that 1/2 is an integer.

This result is FALSE so the assumption was incorrect.

Therefore 2n is NOT ODD so it is EVEN.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #4 2013-10-14 02:58:25

Al-Allo
Member
Registered: 2012-08-23
Posts: 324

AH thank you ! Last question, a real verity will never make any contradictions, right ? (What ever manipulations you do with it )

Last edited by Al-Allo (2013-10-14 03:00:30)

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## #5 2013-10-14 05:10:22

bob bundy
Registered: 2010-06-20
Posts: 8,340

(What ever manipulations you do with it )

Say:  "What ever correct manipulations you do with it " and I'm happy.

In place of "correct" you could also say "valid".

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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## #6 2013-10-14 06:31:10

Al-Allo
Member
Registered: 2012-08-23
Posts: 324

bob bundy wrote:

(What ever manipulations you do with it )

Say:  "What ever correct manipulations you do with it " and I'm happy.

In place of "correct" you could also say "valid".

Bob

AH yes, sorry for the mistake. So yes, any valid manipulation !

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