(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 !
]]>(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
]]>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
]]>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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