Math Is Fun Forum

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Reduction ad Absurdum

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 !

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Reduction ad Absurdum

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.

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Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Reduction ad Absurdum

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Re: Reduction ad Absurdum

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)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Reduction ad Absurdum

(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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Re: Reduction ad Absurdum

(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 !